Distribution Theory

Question 1

Define a probability density function.

Answer 1

Question 2

Define a cumulative distribution function.

Answer 2

Question 3

Define expectation and variance.

Answer 3

Question 4

What is the pdf for the uniform distribution?

Answer 4

Question 5

What is the cdf for the uniform distribution?

Answer 5

Question 6

What is the standard uniform distribution?

Answer 6

a = 0, b = 1 so U ∼ u(0,1)

Question 7

What is the pdf and cdf of the standard uniform distribution?

Answer 7

1. f_u(u) = 1
2. F_u(u) = u

Question 8

How do you derive the cdf of the exponential using the Poisson distribution?

Answer 8

Question 9

What is the pdf of the exponential distribution?

Answer 9

Question 10

How do you find the pdf of the exponential distribution, once you have derived the cdf?

Answer 10

Differentiate it

Question 11

What is a similarity between the exponential and gamma distribution?

Answer 11

• Exponential is the time until the first event occurs
• Gamma is the time until the n^th event occurs

Question 12

How do we derive the cdf of the Gamma distribution?

Answer 12

Question 13

Given the cdf of the Gammas distribution, differentiate it to find the pdf.

Answer 13

Question 14

Define the Gamma distribution.

Answer 14

Question 15

What does Γ(1/2)

Answer 15

sqrt(π)

Question 16

What does Γ(n) equal when n i an integer?

Answer 16

(n-1)!

Question 17

What are three properties of the Gamma distribution?

Answer 17

Question 18

Prove the 𝔼(W) = α/β for the Gamma distribution.

Answer 18

Question 19

What does the pdf and cdf look like for U ∼ u[a,b]?

Answer 19

Question 20

What does the pdf and cdf look like for T ∼ Exp(λ)?

Answer 20

Question 21

What does the pdf and cdf look like for W ∼ Ga(α,β)?

Answer 21

Question 22

If we have a continuous random quantoty X with pdf f_X(x), and y = g(x) what are the two methods to find the distribution and denisty of Y?

Answer 22

1. The cumulative distribution function method
2. The change of variables theorem.

Question 23

What are the two steps in the cumulative distribution method?

Answer 23

Question 24

What pdf f_X(x) and function g(X) do you combine to get the a chi squared distribution?

Answer 24

Z ∼ N(0,1) then Y = Z²

Question 25

How do you derive the cdf of the Chi-squared distribution using the cumulative distribution function method?

Answer 25

Question 26

What are three properties of the chi-squared distribution?

Answer 26

Question 27

What is the univariate transformation theorem?

Answer 27

Question 28

Prove the following theorem.

Answer 28

Question 29

What is the probability integral transform theorem?

Answer 29

Question 30

Prove the following theorem.

Answer 30

Question 31

What is an important use of the probability integral transform theorem?

Answer 31

Inverse sampling - we can generate random numbers from (almost) any distribution by applying an appropiate transformation to a simple standard uniform random numbers

Question 32

What is the algortihm for inverse sampling?

Answer 32

Question 33

Define a joint pdf.

Answer 33

Question 34

Define a joint cdf.

Answer 34

Question 35

Define a marginal pdf.

Answer 35

Question 36

Define a conditional pdf.

Answer 36

Question 37

Define independence between two random quantities.

Answer 37

Question 38

If you have a dsitribution f(x,y) how do you find the marginal distribution fo f_X(x)

Answer 38

Integrate with respect to y

Question 39

What is the multivariate transformation theorem?

Answer 39

Question 40

What is the 100(1-α)% CI for μ using s from a sample?

Answer 40

Question 41

What do we use when we don’t know a value for σ?

Answer 41

s

Question 42

What is the formula for s?

Answer 42

Question 43

When is it wrong to use s?

Answer 43

When s ≠ σ, it will lead to a very wrong confidence interval.

Question 44

What is the Lemma about the following?

Answer 44

Question 45

What is a corollary that links Ẍ and S²?

Answer 45

Question 46

What is the theorem about the Sampling distribution of S²?

Answer 46

Question 47

Prove the following theorem.

Answer 47

Question 48

What is the 100(1 - σ)% confidence interval for σ²?

Answer 48

Question 49

Define a t distribution.

Answer 49

Question 50

What are four properites of the t distribution?

Answer 50

Question 51

What is the theorem about ?

Answer 51

Question 52

Prove the following theorem.

Answer 52

Question 53

What is the confidence interval for μ when σ² is unknown?

Answer 53

Question 54

As n ➝ ∞ what does the t distribution tend to?

Answer 54

The normal