Renormalize files
This commit is contained in:
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Lecture Topic: Probability
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Experiment: An act where the outcome is subject to uncertainty
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Sample space: The set of all possible outcomes of an experiment
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Example:
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- Flip a coin: $S = \{H, T\}$
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- Throw a dice: $S = \{1, 2, 3, 4, 5, 6\}$
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Events: An event is a collection (subset) of outcomes contained in the sample space S
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$$
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S =
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\begin{Bmatrix}
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1,1 & 1,2 & 1,3 & 1,4 & 1,5 & 1,6 \\
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2,1 & 2,2 & 2,3 & 2,4 & 2,5 & 2,6 \\
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3,1 & 3,2 & 3,3 & 3,4 & 3,5 & 3,6 \\
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4,1 & 4,2 & 4,3 & 4,4 & 4,5 & 4,6 \\
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5,1 & 5,2 & 5,3 & 5,4 & 5,5 & 5,6 \\
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6,1 & 6,2 & 6,3 & 6,4 & 6,5 & 6,6
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\end{Bmatrix}
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$$
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A = First and second elements are the same
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$A = \{(1,1) \ (2,2) \ (3,3) \ (4,4) \ (5,5) \ (6,6)\}$
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Union: Union of two events, A and B, ($A \cup B$)
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Compliment: The compliment of an event A, is the set of all outcomes in S that are not in A
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Mutually Exclusive Events: When and B have no common outcomes, they are said to be mutually exclusive or disjoint events, i.e. $P=(A \cap B) = 0$
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Axiom:
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1. For event A, P(A) >= 0
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2. P(S) = 1
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3. (Need to look at slides to correct)
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(a) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
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(b) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
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Proposition:
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1. For event A, P(A) = 1 - P(A')
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2. If and and B are mutually exclusive then $P(A \cap B) = 0$
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Permutation: Any ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects. The number of permutations of size k that can be constructed from n objects is denoted by $P_{k,n}$ and calculated by:
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$$
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P_{k,n} = \frac{n!}{(n-k)!}
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$$
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Combination: Given a set of n, any unordered sub-set of k of the objects is called a combination. The set of combinations of size k that can be formed from n distinct objected will be denoted by $C_{k,n}$ and calculated by:
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$$
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C_{k,n} = \frac{n!}{k!(n-k)!} = \begin{pmatrix}n \\ k \end{pmatrix}
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$$
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(NEED TO REVIEW THIS)
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Example: S = {1, 2, 3, 4, 5}
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$A_1$ = {1,2,3}, $A_2$ = {2,3,5}, $A_3$ = {2,3,1}
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Permutation:
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5! = 120
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(5-3)! = 2! = 2
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120 / 2 = 60
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Combination:
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5!/2!3! = 60/6 = 10
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0! = 1
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5 out of 5
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5!/(5-5)! = 120/1 = 120
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Lecture Topic: Probability
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Experiment: An act where the outcome is subject to uncertainty
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Sample space: The set of all possible outcomes of an experiment
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Example:
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- Flip a coin: $S = \{H, T\}$
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- Throw a dice: $S = \{1, 2, 3, 4, 5, 6\}$
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Events: An event is a collection (subset) of outcomes contained in the sample space S
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$$
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S =
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\begin{Bmatrix}
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1,1 & 1,2 & 1,3 & 1,4 & 1,5 & 1,6 \\
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2,1 & 2,2 & 2,3 & 2,4 & 2,5 & 2,6 \\
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3,1 & 3,2 & 3,3 & 3,4 & 3,5 & 3,6 \\
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4,1 & 4,2 & 4,3 & 4,4 & 4,5 & 4,6 \\
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5,1 & 5,2 & 5,3 & 5,4 & 5,5 & 5,6 \\
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6,1 & 6,2 & 6,3 & 6,4 & 6,5 & 6,6
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\end{Bmatrix}
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$$
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A = First and second elements are the same
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$A = \{(1,1) \ (2,2) \ (3,3) \ (4,4) \ (5,5) \ (6,6)\}$
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Union: Union of two events, A and B, ($A \cup B$)
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Compliment: The compliment of an event A, is the set of all outcomes in S that are not in A
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Mutually Exclusive Events: When and B have no common outcomes, they are said to be mutually exclusive or disjoint events, i.e. $P=(A \cap B) = 0$
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Axiom:
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1. For event A, P(A) >= 0
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2. P(S) = 1
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3. (Need to look at slides to correct)
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(a) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
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(b) If $A_1, A_2 ... A_k$ is a finite collection of mutually exclusive events then: $$P(A_1 \cup A_2 \cup ... \cup A_k) = \sum^{\infty}_{i=1} P(A_i) $$
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Proposition:
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1. For event A, P(A) = 1 - P(A')
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2. If and and B are mutually exclusive then $P(A \cap B) = 0$
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Permutation: Any ordered sequence of k objects taken from a set of n distinct objects is called a permutation of size k of the objects. The number of permutations of size k that can be constructed from n objects is denoted by $P_{k,n}$ and calculated by:
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$$
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P_{k,n} = \frac{n!}{(n-k)!}
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$$
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Combination: Given a set of n, any unordered sub-set of k of the objects is called a combination. The set of combinations of size k that can be formed from n distinct objected will be denoted by $C_{k,n}$ and calculated by:
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$$
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C_{k,n} = \frac{n!}{k!(n-k)!} = \begin{pmatrix}n \\ k \end{pmatrix}
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$$
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(NEED TO REVIEW THIS)
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Example: S = {1, 2, 3, 4, 5}
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$A_1$ = {1,2,3}, $A_2$ = {2,3,5}, $A_3$ = {2,3,1}
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Permutation:
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5! = 120
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(5-3)! = 2! = 2
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120 / 2 = 60
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Combination:
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5!/2!3! = 60/6 = 10
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0! = 1
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5 out of 5
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5!/(5-5)! = 120/1 = 120
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@ -1,2 +1,2 @@
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Lecture Topic:
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Lecture Topic:
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@ -1,5 +1,5 @@
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Lecture Topic: Conditional Probability
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\| (vertical bar) = "given that"
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e.g. P(A|B) = probability of A given that B
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# Question 3
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Lecture Topic: Conditional Probability
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\| (vertical bar) = "given that"
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e.g. P(A|B) = probability of A given that B
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# Question 3
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@ -1,17 +1,17 @@
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Lecture Topic: Random Variable
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Random Variable: Variable with a probability attributed to it
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Regular Variable: Variable that is more intrinsic, like height
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Discrete Variable: Variable where each possible value is a finite set or is an infinite set where each element is it's own distinct element, eg first element, second element etc.
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Probability Mass Function (pmf): The pmf of a discrete rv is defined for every number x by p(x) = P(X=x)
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Mass function criteria:
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- All probabilities have to been between 0 and 1
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- The total probabilities have to equal 1
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Expected Value: The summation of the value and the probability of x over the entire set
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Var?: Variance calculate the expected value but square the variable, then subtract the squared expected value
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Lecture Topic: Random Variable
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Random Variable: Variable with a probability attributed to it
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Regular Variable: Variable that is more intrinsic, like height
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Discrete Variable: Variable where each possible value is a finite set or is an infinite set where each element is it's own distinct element, eg first element, second element etc.
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Probability Mass Function (pmf): The pmf of a discrete rv is defined for every number x by p(x) = P(X=x)
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Mass function criteria:
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- All probabilities have to been between 0 and 1
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- The total probabilities have to equal 1
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Expected Value: The summation of the value and the probability of x over the entire set
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Var?: Variance calculate the expected value but square the variable, then subtract the squared expected value
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@ -1,24 +1,24 @@
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Lecture Topic: Binomial Distribution
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# Requirements of Binomial Experiments
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- (n) independent trials
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- Possible outcomes: success (S) and failure (F)
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- Success probability (p)
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## Formula
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The pmf of binomial rv $X$ depends on two parameters $n$ and $p$. We denote the pmf by $b(x; n,p)$
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$$b(x;n,p) = \{
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\begin{pmatrix}
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n \\
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p \\
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\end{pmatrix}
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p^x(1-p)^{n-x}
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\}$$
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$x = 0, 1, 2, ..., n$
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If X ~ b(x; n,p), then
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1. E(X) = np
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2. V(X) = np(1-p)
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# Examples
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Lecture Topic: Binomial Distribution
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# Requirements of Binomial Experiments
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- (n) independent trials
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- Possible outcomes: success (S) and failure (F)
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- Success probability (p)
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## Formula
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The pmf of binomial rv $X$ depends on two parameters $n$ and $p$. We denote the pmf by $b(x; n,p)$
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$$b(x;n,p) = \{
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\begin{pmatrix}
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n \\
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p \\
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\end{pmatrix}
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p^x(1-p)^{n-x}
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\}$$
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$x = 0, 1, 2, ..., n$
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If X ~ b(x; n,p), then
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1. E(X) = np
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2. V(X) = np(1-p)
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# Examples
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Examples in posted pdf
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@ -1,5 +1,5 @@
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Lecture Topic: Poisson
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Was too interested in configuring neovim, lmao
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Lecture Topic: Poisson
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Was too interested in configuring neovim, lmao
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Look at slides for info, this one seemed important
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@ -1,3 +1,3 @@
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Lecture Topic:
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Lecture Topic:
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Listened to lecturer and didn't take notes
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@ -1,9 +1,9 @@
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Lecture Topic: Continious r.v
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Midterm Review session by math learning center
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Midterm review will be on feb 16, friday
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For evaluating the probability of a distribution of a continious random variable X, you need to integrate the probability
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The probability of a function from - infinity to + infinity must equal 1
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Lecture Topic: Continious r.v
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Midterm Review session by math learning center
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Midterm review will be on feb 16, friday
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For evaluating the probability of a distribution of a continious random variable X, you need to integrate the probability
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The probability of a function from - infinity to + infinity must equal 1
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@ -1,8 +1,8 @@
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Lecture Topic: Some Questions
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Lecture will be short as lecturer needs to leave at 10:00
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First thing to do is to convert x to Z, for solving a standard deviation problem
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$$Z = \frac{x-\mu}{\delta}$$
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Lecture Topic: Some Questions
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Lecture will be short as lecturer needs to leave at 10:00
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First thing to do is to convert x to Z, for solving a standard deviation problem
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$$Z = \frac{x-\mu}{\delta}$$
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Mean is 0 and variance is 1 is the conditions for normal distribution (?) Something standard deviation must be 1 (?) Look into
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@ -1 +1 @@
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Lecture Topic: Binomial Distribution
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Lecture Topic: Binomial Distribution
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@ -1,36 +1,36 @@
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Lecture Topic: Midterm Review 1
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Will be Wednesday, Feb 21, 2024
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Covers Lectures 1-13, Chapters 1-3
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All multiple choice, 18-20 Questions for 50 mins, 16-17 as it is 45 minutes
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45 Minutes, Show up early
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Choose closest answer if answer does not line up
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YOU NEED A CALCULATOR FOR THIS
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For your formula sheet, you can put anything on it, 1 regular size sheet (11x8.5), both sides are allowed to be written on.
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Questions from base theorem will likely take longer than others, so look for questions that are easier to solve first
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## Basic Statistics
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Mean - $\frac{\sum x}{n}$
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Median - Middle Most value, even is the average of two middle values
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Mode - Most frequent value
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Quartile
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- Q1 - 25%, In between 0 and median
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- Q2 = Median
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- Q3 = 75% In Between median and 100
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- Q4 = 100%
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## Variance
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IQR - Inter quartile range - Q3 - Q1
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Variance
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- $\frac{1}{n-1} \sum (x_i - x)^2$ or $\frac{\sum(x_i - x)^2}{n-1}$
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- $\frac{1}{n-1} \sum (x_i^2) - nx^2$ Ask about this, didn't quite catch on board
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- must be positive
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std deviation - Square root of variance, $\sqrt{V(x)}$
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Upper and lower fence - Outlier limits
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- UF = Q3 + 15 IQR Ask about this, could have been 1 times 5, board was messy
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- LF = Q1- 15 IQR Ask about this, could have been 1 times 5, board was messy
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Z score - $\frac{x - \mu}{\sigma}$
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Coefficient of variation / CV = $\frac{\mu}{\sigma} * 100$
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Lecture Topic: Midterm Review 1
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Will be Wednesday, Feb 21, 2024
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Covers Lectures 1-13, Chapters 1-3
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All multiple choice, 18-20 Questions for 50 mins, 16-17 as it is 45 minutes
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45 Minutes, Show up early
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Choose closest answer if answer does not line up
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YOU NEED A CALCULATOR FOR THIS
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For your formula sheet, you can put anything on it, 1 regular size sheet (11x8.5), both sides are allowed to be written on.
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Questions from base theorem will likely take longer than others, so look for questions that are easier to solve first
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## Basic Statistics
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Mean - $\frac{\sum x}{n}$
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Median - Middle Most value, even is the average of two middle values
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Mode - Most frequent value
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Quartile
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- Q1 - 25%, In between 0 and median
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- Q2 = Median
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- Q3 = 75% In Between median and 100
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- Q4 = 100%
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## Variance
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IQR - Inter quartile range - Q3 - Q1
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Variance
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- $\frac{1}{n-1} \sum (x_i - x)^2$ or $\frac{\sum(x_i - x)^2}{n-1}$
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- $\frac{1}{n-1} \sum (x_i^2) - nx^2$ Ask about this, didn't quite catch on board
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- must be positive
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std deviation - Square root of variance, $\sqrt{V(x)}$
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Upper and lower fence - Outlier limits
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- UF = Q3 + 15 IQR Ask about this, could have been 1 times 5, board was messy
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- LF = Q1- 15 IQR Ask about this, could have been 1 times 5, board was messy
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Z score - $\frac{x - \mu}{\sigma}$
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Coefficient of variation / CV = $\frac{\mu}{\sigma} * 100$
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If sigma is not know, replace with S, sample standard deviation
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@ -1,3 +1,3 @@
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Lecture Topic: Exam Review 2
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Lecture Topic: Exam Review 2
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Things to study, distribution, total probability theory
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@ -1,14 +1,14 @@
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Exam Review:
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Calculator Required
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Formula Sheet, 1 page, 2 sided
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Statistical Tables provided
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Everything is multiple choice
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Put scantron sheet in booklet when returning
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Everything can be on the exam, including regression analysis
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Class Wednesday is optional question session
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36-40 Questions expected, stuff after confidence interval may have many multiple choice per question
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When solving a normal distribution problem we convert X to Z
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We do this by subtracting X by mu over sigma
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Exam Review:
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Calculator Required
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Formula Sheet, 1 page, 2 sided
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Statistical Tables provided
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Everything is multiple choice
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Put scantron sheet in booklet when returning
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Everything can be on the exam, including regression analysis
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Class Wednesday is optional question session
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36-40 Questions expected, stuff after confidence interval may have many multiple choice per question
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When solving a normal distribution problem we convert X to Z
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We do this by subtracting X by mu over sigma
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He basically just went over lecture slides again
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