Exam 2 (pt. 1)

Question 1

A random variable X assigns a numerical value X(s) to . . .

Answer 1

each possible outcome s of the experiment

Question 2

The source of the randomness in a random variable is _______________, in which a sample outcome s ∈ S is chosen
according to a ___________

Answer 2

the experiment itself
probability function P

Question 3

A random variable X is said to be discrete if there is a finite list of values a1, a2, . . . , an or an infinite list of values a1, a2, … such that . . .

Answer 3

P(X = aj) ∈ [0, 1]
∑j P(X = aj) = 1

Question 4

If X is a discrete r.v., then the finite or countably infinite set of values x such that P(X = x) > 0 is called the ______ of X.

Answer 4

support

Question 5

The _____________ specifies the probabilities of all events associated with the r.v

Answer 5

distribution of a random variable

Question 6

The probability mass function (PMF) of a discrete r.v. X is the function pX given by _________. Note that this is ________
if x is in the support of X, and _________ otherwise.

Answer 6

pX (x) = P(X = x)

positive, zero

Question 7

The cumulative distribution function (CDF) of an r.v. X is the function F(X) given by __________

Answer 7

F(x) = P(X ≤ x)

Question 8

The expected value of a discrete r.v. X whose distinct possible values are x1, x2, . . ., is defined by . . .

Answer 8

Question 9

The _____________ of X is a weighted average of the possible values that X can take on, weighted by their probabilities

Answer 9

expected value

Question 10

For any r.v.s X, Y and any constant c there are 4 primary manipulations to know for the E(X):
1. E(c) =
2. E(X + Y) =
3. E(X + c) =
4. E(cX) =

Answer 10

Question 11

The variance of an r.v. X is ________

Answer 11

Question 12

For any r.v.s X, Y and any constant c there are 4 primary manipulations to know for the Var(X):
1. Var(X + c) =
2. Var(cX) =
3. Var(X + Y) =
4. Var(X) >= 0 when . . .

Answer 12

Question 13

An r.v. X is said to have the Bernoulli distribution with
parameter p if . . .

Answer 13

P(X = 1) = p and P(X = 0) = 1 − p
where 0 < p < 1

Question 14

A Bernoulli distribution is notated as . . .

Answer 14

X ∼ Bern(p)
where p = P(X = 1)

Question 15

An experiment that can result in either a “success” or a “failure” (but not both) is called a ___________

Answer 15

Bernoulli trial

Question 16

What is the PMF of a Bernoulli distribution?

Answer 16

Question 17

What is the CDF of a Bernoulli distribution?

Answer 17

Question 18

Any r.v. whose possible values are 0 and 1 has a ___________, with p the probability of the r.v. equaling 1. This number p is called the ___________ the distribution

Answer 18

Bern(p) distribution
parameter

Question 19

Let X ∼ Bern(p). Then E(X) = ________

Answer 19

E(X) = 1 · p + 0 · (1 − p) = p

Question 20

Let X ∼ Bern(p). Then Var(X) = ________

Answer 20

Var(X) = p(1 − p)

Question 21

Suppose that n independent Bernoulli trials are performed, each with the same success probability p. Let X be the number of successes. The distribution of X is called the ________ distribution

Answer 21

Binomial

Question 22

A Binomial distribution is notated as . . .

Answer 22

X ∼ Bin(n, p)
parameters n and p, where n is the number of trials and p is the success probability

Question 23

What is the PMF of a Binomial distribution?

Answer 23

Question 24

Let X ∼ Bin(n,p). Then E(X) = ________

Answer 24

E(X) = np

Question 25

Let X ∼ Bin(n,p). Then Var(X) = ________

Answer 25

Var(X) = np(1 − p)

Question 26

If X ∼ Bern(p), then by definition, X ∼ Bin(____)

Answer 26

(1, p)

Question 27

Let X ∼ Bin(n, p). How do we denote the *failure* probability of a Bernoulli trial? Then the distribution becomes _____

Answer 27

q = 1 - p X ∼ Bin(n, q) to denote failures

Question 28

Let X ∼ Bin(4, 1/2). How do you find the CDF F(X= 1.5) such that P(X ≤ 1.5)?

Answer 28

Sum the PMF over all values of the support that are less than or equal to 1.5

Question 29

Let X ∼ Bin(4, 1/2). What is P(X > 1)?

Answer 29

Question 30

Consider a sequence of independent Bernoulli trials, each with the same success probability p ∈ (0, 1), with trials performed until a success occurs. Let X be the number of failures **before** the first successful trial. Then X has the_____________

Answer 30

Geometric distribution

Question 31

A **Geometric distribution** is notated as . . .

Answer 31

X ∼ *Geom*(p) with p as the success probability

Question 32

What is the PMF of a **Geometric distribution** starting at 0?

Answer 32

Question 33

Let X ∼ Geom(p). Then E(X) = ________

Answer 33

E(X) = (1 - p)/p

Question 34

Let X ∼ Geom(p). Then Var(X) = ________

Answer 34

Var(X) = (1 - p)/p^2

Question 35

In a sequence of independent Bernoulli trials with success probability p, if X is the number of failures before the rth success, then X is said to have the ______

Answer 35

Negative Binomial distribution

Question 36

A **Negative binomial distribution** is notated as . . .

Answer 36

X ∼ *NBin*(r, p) where r is the desired number success and p is the probability of success

Question 37

What is the PMF of a **Negative binomial distribution**?

Answer 37

Question 38

Let X ∼ NBin(r,p). Then E(X) = ________

Answer 38

E(X) = (1 − p)/p

Question 39

Let X ∼ NBin(r,p). Then Var(X) = ________

Answer 39

Var(X) = r(1 - p)/p^2

Question 40

An r.v. X has the __________ with parameter λ if it explains the distribution of . . . 1. the number of successes in a particular region or _________ 2. a large number of trials, each with a ___________.

Answer 40

Poisson distribution interval of time small probability of success

Question 41

What is the PMF of a **Poisson distribution**?

Answer 41

Question 42

A **Poisson distribution** is notated as . . .

Answer 42

X ~ *Pois*( λ) Where parameter λ is interpreted as the rate of occurrence of these rare events

Question 43

Let X ~ Pois(λ). Then E(X) = _________

Answer 43

E(X) = λ

Question 44

Let X ~ Pois(λ). Then Var(X) = _________

Answer 44

Var(X) = λ

Question 45

The Poisson paradigm is also called the law of rare events. The interpretation of "rare" is that the _____ are small, not that _______ is small.

Answer 45

p λ

Question 46

Given X ~ Bin(n,p) where n is large, p is small, and np is moderate, we can approximate the Bin(n, p) PMF by the _____________

Answer 46

Pois(np) PMF

Question 47

Let C be a finite, nonempty set of numbers. Choose one of these numbers uniformly at random (i.e., all values in C are equally likely). Call the chosen number X. Then X is said to have the _________________

Answer 47

Discrete Uniform distribution

Question 48

A **Discrete Uniform Distribution** is notated as . . .

Answer 48

X ~ *Unif*(C) Where parameter C is the finite, nonempty set of numbers

Question 49

What is the PMF of a **Discrete Uniform** distribution?

Answer 49

Question 50

Let X ~ Unif(C). Then E(X) = __________

Answer 50

(a + b)/2 where a is the minimum of the set C and b is the maximum

Question 51

Let X ~ Unif(C). Then Var(X) = __________

Answer 51

where a is the minimum of the set C and b is the maximum

Question 52

For an experiment with sample space S, an r.v. X, and a function g : R → R, g(X) is the r.v. that maps s to _____________

Answer 52

g(X(s)) for all s ∈ S

Question 53

Given a discrete r.v. X with a known PMF, how can we find the PMF of Y = g(X)?

Answer 53

Question 54

Random variables X and Y are said to be independent if . . .

Answer 54

P(X ≤ x, Y ≤ y) = P(X ≤ x)P(Y ≤ y) or for discrete cases P(X = x, Y = y) = P(X = x)P(Y = y)

Question 55

If X ∼ Bin(n, p), Y ∼ Bin(m, p), and X is independent of Y, then . . .

Answer 55

X + Y ∼ Bin(n + m, p)

Question 56

Random variables X and Y are said to be **conditionally independent** given an r.v. Z if for all x, y ∈ R and all z in the support of Z, if . . .

Answer 56

P(X ≤ x, Y ≤ y | Z = z) = P(X ≤ x | Z = z)P(Y ≤ y | Z = z) or for discrete cases P(X = x, Y = y | Z = z) = P(X = x | Z = z)P(Y = y | Z = z)

Question 57

For any discrete r.v.s X and Z, the function P(X = x|Z = z), when considered as a function of x for fixed z, is called the _________________ of X given Z = z.

Answer 57

conditional PMF