Exam 2 (pt. 2)

Question 1

An r.v. has a continuous distribution if its CDF is ___________.
We also allow there to be endpoints (or finitely many points) where the CDF is continuous but not differentiable, as long as . . .

Answer 1

differentiable
the CDF is differentiable everywhere else

Question 2

Let X be a continuous r.v. with PDF f . Then the CDF of X is given by . . .

Answer 2

Question 3

Probabilities for continuous random variables are specified by__________. This leads us to the special rule that P(X = x) = _______ for continuous random variables

Answer 3

the area under a curve
0

Question 4

For sets of the form [a, b], (a, b], [a, b), (a, b), the CDF is dereived from the PMF by . . .

Answer 4

Question 5

To get a desired probability for a continuous r.v. you . . .

Answer 5

integrate the PDF over the desired range

Question 6

The PDF f of a continuous r.v. must satisfy the following two criteria:

Answer 6

Question 7

The expected value (also called the expectation or mean) of a continuous r.v. X with PDF f is . . .

Answer 7

Question 8

If X is a continuous r.v. with PDF f and g is a function from R to R, then E(g(X))= ___

Answer 8

Question 9

To go from PDF to CDF of a continuous distribution you _______

Answer 9

Integrate f(x)

Question 10

What is the power rule for differentiating a function?

Answer 10

Question 11

To go from a CDF to a PDF of a continuous distribution you _______

Answer 11

differentiate F(x)

Question 12

What is the power rule for integrating a function?

Answer 12

Question 13

To find the probability of P(a < x < b) you ____________ the _________ on (a,b)

Answer 13

integrate, PDF

Question 14

A continuous r.v. U is said to have the Uniform distribution on the interval (a, b) if its PDF is . . .

Answer 14

Question 15

The CDF of the continuous uniform is . . .

Answer 15

Question 16

The uniform distribution is denoted as ________. The standard uniform is _______

Answer 16

U ~ Unif(a , b)
U ~ Unif(0 , 1)

Question 17

What is E(U) of U ~ Unif(a , b)?

Answer 17

E(U) = (a + b)/2

Question 18

What is Var(U) of U ~ Unif(a , b)?

Answer 18

(b - a)^2/12

Question 19

A continuous r.v. X is said to have the Exponential distribution with parameter λ > 0 if its PDF is . . .

Answer 19

Question 20

A continuous r.v. X is said to have the Exponential distribution with parameter λ > 0 if its CDF is . . .

Answer 20

Question 21

The exponential distribution is denoted as . . .

Answer 21

X ~ Exp(λ)

Question 22

If X ~ Exp(1), then Y ~ λ^-1X ~ ________.
What would the converse be?

Answer 22

Question 23

What is E(X) and Var(X) given that X ~ Exp(λ)?

Answer 23

Question 24

A continuous distribution is said to have the memoryless property if a random variable X from that distribution satisfies . . .

Answer 24

Question 25

The memoryless property is a very special property of the Exponential distribution because . . .

Answer 25

no other continuous distribution on (0, ∞) is memoryless

Question 26

A continuous r.v. Z is said to have the **standard** Normal distribution if its PDF ϕ is given by . . .

Answer 26

Question 27

How is the standard, normal distribution notated?

Answer 27

Z ~ *N*(0,1)

Question 28

What is the CDF of the standard, normal distribution?

Answer 28

Question 29

the **central limit theorem** says that under very weak assumptions, the sum of a large number of i.i.d. random variables has an . . .

Answer 29

approximately Normal distribution, regardless of the distribution of the individual r.v.s.

Question 30

What are the three important symmetry properties that can be deduced from the standard Normal PDF and CDF?

Answer 30

Question 31

For a standard, normal distribution, the E(Z) = _______

Answer 31

Question 32

For a standard, normal distribution, the Var(Z) = ________

Answer 32

Question 33

If Z ∼ N(0, 1) then X = μ + σZ has . . .

Answer 33

a normal distribution with mean μ and variance σ^2, for any μ ∈ R, and σ^2 with σ > 0.

Question 34

By linearity, E(X) of a normal distribution = _________

Answer 34

Question 35

By linearity, Var(X) of a normal distribution = _________

Answer 35

Question 36

For X ∼ N(μ, σ2) the standardized version of X is given by . . .

Answer 36

Question 37

For X ∼ N(μ, σ2) the standardized CDF of X is . . .

Answer 37

Question 38

For X ∼ N(μ, σ2) the standardized PDF of X is . . .

Answer 38

Question 39

Let X ∼ N(−1, 4). What is P(|X| < 3), exactly in terms of Φ?

Answer 39

Question 40

If X ∼ N(μ, σ2) then . . . 1. P(|X − μ| < σ) ≈ 2. P(|X − μ| < 2σ) ≈ 3. P(|X − μ| < 3σ) ≈ This is commonly known as _____ rule

Answer 40

1. P(|X − μ| < σ) ≈ 0.68 2. P(|X − μ| < 2σ) ≈ 0.95 3. P(|X − μ| < 3σ) ≈ 0.997 68-95-99.7% rule

Question 41

Let X ∼ N(−1, 4). What is P(|X| < 3), approximately?

Answer 41

Question 42

Given that X ~ *Exp*(λ) what is a rate parameter versus a scale parameter?

Answer 42

Rate parameter = λ Scale Parameter = 1/λ

Question 43

What is P(c < x < d) for X~Unif(a,b)?

Answer 43

(d-c/b-a)

Question 44

How do you find percentile of a normal distribution?

Answer 44

X = μ + (z x σ) where z corresponds to the Z score (X − μ /σ) of the desired percentile