week 5 flashcards

Question 1

What is a random variable (RV)?

Answer 1

A function that assigns numerical values to outcomes of a random process.

Question 2

Expected Value (Mean) of X?

Answer 2

E[X] = Σ x P(x)

Question 3

Variance Formula?

Answer 3

Var[X] = E[(X - μ)^2] p[x]

Question 4

Standard Deviation Formula?

Answer 4

SD[X] = √Var[X]

Question 5

What happens when you add/subtract a constant C to X?

Answer 5

Mean shifts: E[X + C] = E[X] + C, but variance remains the same.

Question 6

What happens when you multiply X by a constant C?

Answer 6

Mean scales: E[aX] = aE[X], Variance scales: Var[aX] = a²Var[X]

Question 7

What are Bernoulli trials?

Answer 7

Repeated experiments with 2 outcomes (success/failure).

Question 8

Geometric Distribution Formula?

Answer 8

P(X = k) = (1 - p)^(k - 1) p

Question 9

Expected value of a Geometric RV?

Answer 9

E[X] = 1/p

Question 10

Variance of a Geometric RV?

Answer 10

Var[X] = (1 - p) / p²

Question 11

Binomial Distribution Formula?

Answer 11

P(X = k) = (n choose k) p^k (1 - p)^(n - k)

Question 12

Expected value of Binomial RV?

Answer 12

E[X] = np

Question 13

Variance of Binomial RV?

Answer 13

Var[X] = np(1 - p)

Question 14

Poisson Distribution Formula?

Answer 14

P(X = n) = (e^(-λ) λ^n) / n!

Question 15

Expected value of Poisson RV?

Answer 15

E[X] = λ

Question 16

Variance of Poisson RV?

Answer 16

Var[X] = λ

Question 17

What is the Normal Distribution?

Answer 17

A bell-shaped curve defined by mean μ and standard deviation σ.

Question 18

Standard Normal Distribution Formula?

Answer 18

Z = (X - μ) / σ

Question 19

What is the Exponential Distribution?

Answer 19

Models time until the next event in a Poisson process.

Question 20

Exponential Distribution Formula?

Answer 20

P(X ≤ t) = 1 - e^(-λ t)

Question 21

Sum of two independent RVs?

Answer 21

E[X + Y] = E[X] + E[Y], Var[X + Y] = Var[X] + Var[Y]

Question 22

What happens when you subtract two RVs?

Answer 22

Expected value subtracts, but variance always adds: Var[X - Y] = Var[X] + Var[Y]

Question 23

Probability of rolling exactly 3 sixes in 10 rolls?

Answer 23

Use Binomial: P(X = k) = (n choose k) p^k (1 - p)^(n - k)

Question 24

Website gets 4 visitors per minute. Probability of exactly 5 visitors in a minute?

Answer 24

Use Poisson: P(X = 5) = (e^(-4) 4^5) / 5!

Question 25

If X is multiplied by 3, how does variance change?

Answer 25

Var[3X] = 3² Var[X] = 9Var[X]

Question 26

What does variance measure?

Answer 26

Spread of values around the mean.

Question 27

When do you use a Binomial model?

Answer 27

When counting successes in fixed trials.

Question 28

When do you use a Poisson model?

Answer 28

When counting events over time/space.

Question 29

What does the standard deviation tell us?

Answer 29

The average deviation from the mean.

Question 30

What does the Normal model assume?

Answer 30

Data is bell-shaped and symmetric.

Question 31

Why does a Poisson model work for rare events?

Answer 31

Events happen randomly over time/space.

Question 32

How does adding a constant affect variance?

Answer 32

It doesn’t change variance, only shifts the mean.

Question 33

What is the rule for variances of independent sums?

Answer 33

Always add variances, even for subtraction.

Question 34

What happens to a Poisson process when λ increases?

Answer 34

Events happen more frequently.