Probabilities, Distributions, Expected Value, & IID variables

What does mutually exclusive events mean?

2 events can’t happen at the same time

What is the conditionally probability of B given A

What does independent events mean, and what’s it notation

• Probability of second event remains the same even if you know the first event happened

  • A and B are independent if: P(B|A) = P(B)

What does A U B mean

Define new event A or B

• P(A) or P(B) or both

• P(A) + P(B) - P(both)

What does A ‘upside down U’ B mean

Define new event A and B

What does Discrete Uniform Distribution mean

• each possible outcome has the same probability (1/n)

What’s bernouli distribution

probability with 2 outcomes (success & failure) → (1 & 0) → “p” usually means success, and

(1 - p) means failure

What’s a binomial distribution

•  a sequence of n independent Bernoulli trials

What’s a Hypergeometric Distribution

•  probability of k success in n draws without replacement

What is the equation for a hypergeometric distribution

• G: tickets marked 1

• K: number of samples drawn with k tickets marked 1

• (N - G): tickets marked 0 in the box

• (N n) – denominator: total simple random samples of size n that can be drawn from pool N

What are some key properties of Variance

• Can never be negative

• Adding constants doesn’t change anything

• Multiply every outcome by constant a → variance inc by a^2

  • e.g) Var(aX) → a^2 Var(x)

• Var( X + Y) = Var(X) + Var(Y)

• Var(X - Y) = Var(X) + Var(Y)

What does expected value tell you?

Tells you where the center of your data is

What does the Central Limit Thm tell you?

Take many samples from a population and calculate each mean —> as samples increase —> distribution normal

Can you give me the coordinates, and probabilities under the Gaussian Distribution

What is the expected value of the sample mean

What is the variance of sample mean