Probability Mass Function (PMF) Practice Assessment — Questions and Answers

Question 1: What does it mean for a PMF to be uniform on {1, 2, 3, 4, 5}?

The PMF values decrease uniformly
Each outcome has probability proportional to k
The CDF is linear everywhere
Each outcome has P(X=k) = 1/5 = 0.2 (Correct answer)

Correct answer: Each outcome has P(X=k) = 1/5 = 0.2

A uniform PMF assigns equal probability 1/n to each of the n outcomes; here, P(X=k) = 1/5 = 0.2 for each k.

Question 2: For a PMF, what value minimizes E[(X - c)²] over all constants c?

c = Var(X)
c = mode(X)
c = median(X)
c = E[X] (the mean) (Correct answer)

Correct answer: c = E[X] (the mean)

The mean E[X] minimizes the mean squared error E[(X-c)²], a fundamental property of expectation.

Question 3: What does Chebyshev's inequality state for a PMF with mean μ and variance σ²?

P(|X - μ| ≤ kσ) ≤ 1/k²
P(X ≥ μ) ≥ 0.5
P(X ≤ μ + kσ) ≥ 1 - 1/k
P(|X - μ| ≥ kσ) ≤ 1/k² for any k > 1 (Correct answer)

Correct answer: P(|X - μ| ≥ kσ) ≤ 1/k² for any k > 1

Chebyshev's inequality states P(|X-μ| ≥ kσ) ≤ 1/k², bounding the probability of being far from the mean using only mean and variance.

Question 4: If X counts failures before the r-th success and Y counts total trials, how are X and Y related?

Y = X - r
Y = X + r (Correct answer)
Y = r × X
Y = X

Correct answer: Y = X + r

Total trials Y = failures X + successes r, so Y = X + r.

Question 5: In Binomial Distribution, each trial must be independent.

Not defined
None of the above
TRUE (Correct answer)
FALSE

Correct answer: TRUE

In Binomial Distribution, each trial must be independent. This means that the outcome of one trial does not affect the outcome of any other trial. Each trial is considered to be separate and unrelated to the others. This is a fundamental assumption in the binomial distribution, as it allows for the calculation of probabilities based on the number of successes and failures in a fixed number of independent trials.

Question 6: As r → ∞ with mean held constant, what distribution does the Negative Binomial approach?

Geometric distribution
Normal distribution
Poisson distribution (Correct answer)
Binomial distribution

Correct answer: Poisson distribution

As r → ∞ with mean μ = r(1-p)/p held fixed, the Negative Binomial converges to the Poisson distribution.

Question 7: For a PMF with values {1,2,3,4}, if F(3) = 0.8 and F(4) = 1, what is P(X=4)?

0.4
0.8
0.2 (Correct answer)
1.0

Correct answer: 0.2

P(X=4) = F(4) - F(3) = 1.0 - 0.8 = 0.2.

Question 8: The Negative Binomial distribution is overdispersed relative to the Poisson. What does this mean?

They have equal mean and variance
Its mean exceeds its variance
It is symmetric like the Poisson
Its variance exceeds its mean (Correct answer)

Correct answer: Its variance exceeds its mean

Overdispersion means Var(X) > E[X], which the Negative Binomial exhibits, making it useful when Poisson's equal mean-variance assumption is violated.

Question 9: If X and Y are discrete random variables, E[g(X, Y)] is computed as:

g(E[X], E[Y])
∑_x ∑_y g(x, y) · p(x, y) (Correct answer)
∑_x g(x) · ∑_y g(y)
∑_x ∑_y g(x, y) · p_X(x) · p_Y(y)

Correct answer: ∑_x ∑_y g(x, y) · p(x, y)

The expected value of any function g(X, Y) is ∑_x ∑_y g(x, y) · p(x, y), weighting each value by its joint probability.

Question 10: An inspector needs to find 3 defective items. Each item has a 20% chance of being defective. What distribution models the number of items inspected until the 3rd defect?

Binomial with n=3, p=0.2
Negative Binomial with r=3, p=0.2 (Correct answer)
Poisson with λ=0.6
Geometric with p=0.2

Correct answer: Negative Binomial with r=3, p=0.2

Waiting for a specified number (r=3) of successes uses the Negative Binomial distribution.

Question 11: Which is a type of random variables?

Normal
Ordinal
Linear
Continuous (Correct answer)

Correct answer: Continuous

A continuous random variable is a type of random variable that can take on any value within a certain range or interval. It can have an infinite number of possible outcomes and is typically associated with measurements or quantities that can be expressed as real numbers. Examples of continuous random variables include height, weight, time, and temperature. In contrast, other options like linear, normal, and ordinal are not types of random variables but rather describe different characteristics or properties of random variables.

Question 12: How does increasing p affect the Geometric PMF?

The distribution becomes more spread out
The mean increases
The distribution becomes more concentrated near k=1 (Correct answer)
The variance increases

Correct answer: The distribution becomes more concentrated near k=1

Higher p means success is more likely quickly, so the distribution concentrates near smaller values of k.

Question 13: Which of these restricts the measurement of a continuous variable?

Figure
Number
Accuracy (Correct answer)
Place

Correct answer: Accuracy

Accuracy restricts the measurement of a continuous variable because accuracy refers to the degree of closeness between a measured value and the true value of the variable. In the context of measurement, accuracy determines how precise and reliable the measurement is. Therefore, if the accuracy is low, it means that the measurement may have a larger margin of error, limiting the ability to accurately measure and quantify the continuous variable.

Question 14: What does the Geometric distribution model?

The number of trials until the first success (Correct answer)
The time between two Poisson events
The total number of successes in n trials
The number of failures before the k-th success

Correct answer: The number of trials until the first success

The Geometric distribution models the number of independent Bernoulli trials needed to achieve the first success.

Question 15: What is the support of the Geometric distribution (number-of-trials version)?

{1, 2, 3, 4, …} (Correct answer)
{0, 1, 2, 3, …}
{0, 1}
{1, 2, …, n}

Correct answer: {1, 2, 3, 4, …}

Since at least one trial is always needed, the support starts at 1 and extends to infinity.

Question 16: Random variables that can take on any of the countless number of values in an interval are called...

Open
Closed
Continuous (Correct answer)
Discrete

Correct answer: Continuous

Random variables that can take on any of the countless number of values in an interval are called continuous. This means that the variable can take on any value within a range, including fractions and decimals. Unlike discrete variables, which can only take on specific, separate values, continuous variables have an infinite number of possible values. Closed and open are not appropriate terms to describe random variables, as they refer to intervals rather than the variables themselves.

Question 17: What is the support of the Negative Binomial (total trials until r-th success)?

{0, 1, 2, …}
{1, 2, …, r}
{r, r+1, r+2, …} (Correct answer)
{r, r+1, …, n}

Correct answer: {r, r+1, r+2, …}

The minimum number of trials to get r successes is r (all trials are successes), so the support starts at k = r.

Question 18: The finite population correction factor in the Hypergeometric variance is (N-n)/(N-1). What happens to this factor as n approaches N?

It approaches 0, reducing the variance (Correct answer)
It approaches 1, giving the Binomial variance
It becomes negative
It approaches N

Correct answer: It approaches 0, reducing the variance

As n → N, (N-n)/(N-1) → 0, meaning the variance shrinks to zero because when you sample the entire population, there is no randomness.

Question 19: For a PMF with P(X=0)=0.5, P(X=2)=0.3, P(X=4)=0.2, what is Var(X)?

2 (Correct answer)
2.4
3.2
1.96

Correct answer: 2

E[X]=0(0.5)+2(0.3)+4(0.2)=0+0.6+0.8=1.4; E[X²]=0(0.5)+4(0.3)+16(0.2)=0+1.2+3.2=4.4; Var=4.4-1.96=2.44≈2.44, approximately 2.

Question 20: Given joint PMF p(1,1)=0.1, p(1,2)=0.2, p(2,1)=0.3, p(2,2)=0.4, what is the marginal PMF p_X(1)?

0.1
0.4
0.7
0.3 (Correct answer)

Correct answer: 0.3

p_X(1) = p(1,1) + p(1,2) = 0.1 + 0.2 = 0.3, summing over all values of Y.

Question 21: The marginal PMF of X is obtained from the joint PMF p(x, y) by:

Multiplying p(x, y) by p(y) for each y
Summing p(x, y) over all values of y (Correct answer)
Dividing p(x, y) by p(y) for each y
Summing p(x, y) over all values of x

Correct answer: Summing p(x, y) over all values of y

The marginal PMF p_X(x) = ∑_y p(x, y) is obtained by summing the joint PMF over all values of the other variable Y.

Question 22: What is the mean of a Hypergeometric distribution with parameters N, K, n?

nK
n × p
K/N
n × K/N (Correct answer)

Correct answer: n × K/N

E[X] = n × K/N, which is the sample size times the population proportion of successes.

Question 23: Let X and Y have joint PMF p(x, y) = 1/6 for (x, y) ∈ {(1,1),(1,2),(2,1),(2,2),(3,1),(3,2)}. What is p_Y(1)?

1/2 (Correct answer)
1/3
2/3
1/6

Correct answer: 1/2

p_Y(1) = p(1,1) + p(2,1) + p(3,1) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2.

Question 24: A lot of 20 items contains 4 defectives. A sample of 5 is drawn without replacement. What is E[X], the expected number of defectives?

0.8
4
1 (Correct answer)
2

Correct answer: 1

E[X] = n × K/N = 5 × 4/20 = 5 × 0.2 = 1.

Question 25: Suppose p(x, y) = c · x · y for x ∈ {1, 2} and y ∈ {1, 2}, and 0 otherwise. What is the value of c?

1/6
1/3
1/4
1/9 (Correct answer)

Correct answer: 1/9

Summing c·x·y over all pairs: c(1·1 + 1·2 + 2·1 + 2·2) = c(1+2+2+4) = 9c = 1, so c = 1/9.

Question 26: A random variable can also be referred to as...

Open
Clear
Branched
Stochastic (Correct answer)

Correct answer: Stochastic

A random variable can also be referred to as "stochastic" because it represents a variable whose outcome is determined by chance or probability. Stochastic variables are used in probability theory and statistics to model and analyze uncertain events or processes. They can take on different values with varying probabilities, and their behavior is often described using probability distributions.

Question 27: If a variable can take on any value between two specified values, it is called...

Continuous variable (Correct answer)
Line variable
Discrete variable
Real

Correct answer: Continuous variable

A continuous variable is a variable that can take on any value within a specified range. Unlike discrete variables, which can only take on specific values, a continuous variable can take on any value within a range, including decimal values. In this case, the variable can take on any value between two specified values, indicating that it is a continuous variable.

Question 28: For the joint PMF p(x, y) = 1/9 · x · y with x, y ∈ {1, 2}, are X and Y independent?

No, because E[XY] ≠ E[X]E[Y]
Yes, because the joint PMF is symmetric
No, because the marginals are not uniform
Yes, because p(x, y) = p_X(x) · p_Y(y) for all x, y (Correct answer)

Correct answer: Yes, because p(x, y) = p_X(x) · p_Y(y) for all x, y

With p_X(1)=1/3, p_X(2)=2/3, p_Y(1)=1/3, p_Y(2)=2/3, we verify p(x,y) = p_X(x)·p_Y(y) holds for all pairs, confirming independence.

Question 29: Two discrete random variables X and Y have Cov(X, Y) = 0. This means:

The joint PMF must factor as a product of marginals
X and Y are uncorrelated, but not necessarily independent (Correct answer)
X and Y are always independent
X and Y must have the same distribution

Correct answer: X and Y are uncorrelated, but not necessarily independent

Zero covariance implies uncorrelatedness, but independence is a stronger condition; uncorrelated variables can still be dependent.

Question 30: A Geometric PMF decreases monotonically. What does this imply about most likely outcome?

All outcomes are equally likely
The median equals the mean
Large k values have higher probability
The first trial (k=1) always has the highest probability of success (Correct answer)

Correct answer: The first trial (k=1) always has the highest probability of success

Since P(X=k) = (1-p)^(k-1) × p decreases as k increases, the most probable outcome is always k=1.

Question 31: A store averages 3 customer complaints per day. Using the Poisson PMF, what is P(X=3)?

e^-3 ≈ 0.050
(e^-3 × 9) / 2 ≈ 0.224
(e^-3 × 27) / 6 ≈ 0.224 (Correct answer)
3 × e^-3 ≈ 0.149

Correct answer: (e^-3 × 27) / 6 ≈ 0.224

P(X=3) = (e^-3 × 3^3) / 3! = (e^-3 × 27) / 6 ≈ 0.224.

Question 32: How does Var(aX + b) relate to Var(X) for constants a and b?

Var(aX+b) = a²×Var(X) + b²
Var(aX+b) = a×Var(X) + b
Var(aX+b) = a²×Var(X) (Correct answer)
Var(aX+b) = Var(X)

Correct answer: Var(aX+b) = a²×Var(X)

Var(aX+b) = a²×Var(X) because adding a constant b does not change spread, and scaling by a scales variance by a².

Question 33: Variables which cannot take any value at all are...

Ordinal
Nonrandom (Correct answer)
Calling
Ordinary

Correct answer: Nonrandom

Nonrandom variables are those that do not have any variation or variability in their values. In other words, these variables cannot take on different values and remain constant. They are fixed and unchanging. In contrast, random variables can take on different values based on chance or probability. Therefore, nonrandom variables cannot take any value at all, as they are not subject to variation or randomness.

Question 34: If X ~ Poisson(λ=4), what is P(X=1)?

8e^-4 ≈ 0.147
4e^-4 ≈ 0.073 (Correct answer)
4e^4
e^-4 ≈ 0.018

Correct answer: 4e^-4 ≈ 0.073

P(X=1) = (e^-4 × 4^1) / 1! = 4e^-4 ≈ 0.073.

Question 35: The law of total probability for PMFs states that p_X(x) can be written as:

p_X(x) = p(x, y) · p_Y(y)
p_X(x) = ∏_y p(x | y)
p_X(x) = ∑_y p(x | y) · p_Y(y) (Correct answer)
p_X(x) = p(x, y) / p_Y(y)

Correct answer: p_X(x) = ∑_y p(x | y) · p_Y(y)

By the law of total probability, p_X(x) = ∑_y p(x | y) · p_Y(y), which expands the marginal using conditional probabilities.

Question 36: For a discrete PMF, what type of function is the CDF?

A smooth continuous function
A linear function
A differentiable function
A right-continuous step function (Correct answer)

Correct answer: A right-continuous step function

The CDF of a discrete distribution is a step function that jumps at each mass point and is right-continuous.

Question 37: Which formula correctly expresses the Poisson PMF for k occurrences with rate λ?

P(X=k) = (e^-λ × λ^k) / k! (Correct answer)
P(X=k) = (e^λ × k^λ) / k!
P(X=k) = (λ^k × k!) / e^λ
P(X=k) = λ^k / (e^λ × k)

Correct answer: P(X=k) = (e^-λ × λ^k) / k!

The Poisson PMF is P(X=k) = (e^-λ × λ^k) / k! where λ is the average rate and k is a non-negative integer.

Question 38: What is the standard deviation of X if Var(X) = 9?

9
3 (Correct answer)
4.5
81

Correct answer: 3

SD(X) = √Var(X) = √9 = 3.

Question 39: A variable whose possible values are numerical outcomes of a random phenomenon is termed...

Closed variable
Random variable (Correct answer)
Clear variable
Open variable

Correct answer: Random variable

A random variable is a term used to describe a variable that can take on different numerical outcomes as a result of a random or uncertain event or phenomenon. This means that the values of the variable are not predetermined or fixed, but rather depend on the outcome of the random process. Therefore, the correct answer is "Random variable."

Question 40: Random variables that can assume only a countable number of values are called...

Discrete (Correct answer)
Closed
Continuous
Open

Correct answer: Discrete

Random variables that can assume only a countable number of values are called discrete. Discrete random variables have specific, separate, and distinct values. They can be expressed as whole numbers or a finite set of values. Unlike continuous random variables, discrete random variables cannot take on any value within a given range.

Probability Mass Function (PMF) Practice Assessment

A statistics assessment covering discrete probability distributions and probability mass functions, including geometric, hypergeometric, and joint/marginal PMF topics used in probability theory and applied statistics courses.

Questions

60 min

Time Limit

70%

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