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

Question 1: If X ~ Poisson(λ=2), what is P(X=0)?

0.5
e^-2 ≈ 0.135 (Correct answer)
1 - e^-2 ≈ 0.865
2e^-2 ≈ 0.271

Correct answer: e^-2 ≈ 0.135

P(X=0) = (e^-2 × 2^0) / 0! = e^-2 ≈ 0.135.

Question 2: The qualities of discrete data can be:

Both
None
Measured
Counted (Correct answer)

Correct answer: Counted

Discrete data refers to data that can only take on specific values and cannot be measured on a continuous scale. It can only be counted or enumerated. This means that the values of discrete data can be expressed as whole numbers or integers, such as the number of students in a class or the number of cars in a parking lot. Therefore, the correct answer is "Counted."

Question 3: If a variable can take on any value between two specified values, it is called...

Discrete variable
Real
Continuous variable (Correct answer)
Line variable

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 4: For a joint PMF p(x, y), the covariance Cov(X, Y) is computed as:

E[(X + Y)²] − (E[X] + E[Y])²
Var(X) + Var(Y)
E[XY] − E[X]·E[Y] (Correct answer)
E[X²Y²] − E[X]²E[Y]²

Correct answer: E[XY] − E[X]·E[Y]

Cov(X, Y) = E[XY] − E[X]·E[Y], measuring how much X and Y vary together relative to their individual means.

Question 5: Suppose p(x, y) = c · x · y for x ∈ {1, 2} and y ∈ {1, 2}, and 0 otherwise. What is the value of c?

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

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 6: A researcher needs 5 volunteer sign-ups. Each person contacted has a 30% chance of volunteering. What is E[total people contacted]?

0.7/0.3 × 5 ≈ 11.67
5/0.3 ≈ 16.67 (Correct answer)
5 × 0.7 = 3.5
5 × 0.3 = 1.5

Correct answer: 5/0.3 ≈ 16.67

For total trials until r successes, E[X] = r/p = 5/0.3 ≈ 16.67 people need to be contacted.

Question 7: What is the standard deviation of X if Var(X) = 9?

9
4.5
3 (Correct answer)
81

Correct answer: 3

SD(X) = √Var(X) = √9 = 3.

Question 8: Which of these restricts the measurement of a continuous variable?

Number
Place
Figure
Accuracy (Correct answer)

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 9: What is the relationship between PMF and the probability of X falling in any set A?

P(X ∈ A) = P(X=max(A))
P(X ∈ A) = Σ P(X=x) for all x in A (Correct answer)
P(X ∈ A) = ∫_A P(X=x) dx
P(X ∈ A) = |A| × P(X=x)

Correct answer: P(X ∈ A) = Σ P(X=x) for all x in A

For discrete X, the probability of any event is the sum of the PMF over all outcomes in that event.

Question 10: Given p(x, y) for x ∈ {0, 1} and y ∈ {0, 1, 2}, how many distinct probability values define the joint PMF?

5
6 (Correct answer)
2
3

Correct answer: 6

The joint PMF is defined over 2 × 3 = 6 pairs (x, y), each requiring a probability value, though the values must sum to 1.

Question 11: What is the sum of all probabilities P(X=k) for k=0,1,2,… in a Poisson distribution?

λ
1 (Correct answer)
e^λ
e^-λ

Correct answer: 1

Like all valid PMFs, the Poisson probabilities sum to 1 over all possible values.

Question 12: Which condition must be met for the Poisson distribution to be a valid model?

Each trial must have exactly two outcomes
The probability of success must exceed 0.5
Events must occur independently at a constant average rate (Correct answer)
The number of trials must be fixed

Correct answer: Events must occur independently at a constant average rate

The Poisson model requires events to be independent and to occur at a constant average rate within a fixed interval.

Question 13: For a discrete PMF, what type of function is the CDF?

A differentiable function
A smooth continuous function
A linear 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 14: Which is a type of random variables?

Linear
Continuous (Correct answer)
Normal
Ordinal

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 15: For a valid PMF, can P(X = x₀) = 0 for some specific value x₀ in the support?

Only if there are infinitely many values
Yes, a PMF can assign zero probability to values in its defined domain (Correct answer)
Only if x₀ is not in the support
No, all defined values must have positive probability

Correct answer: Yes, a PMF can assign zero probability to values in its defined domain

P(X=x) ≥ 0 allows values of zero; only values in the support (where P > 0) are typically listed.

Question 16: Which property makes the Geometric distribution unique among discrete distributions?

Memoryless property (Correct answer)
Both mean and variance equal λ
Sum of two Geometric variables is Geometric
Symmetry around the mean

Correct answer: Memoryless property

The Geometric distribution is the only discrete distribution with the memoryless property: past failures do not affect future probabilities.

Question 17: For two discrete random variables X and Y, the joint PMF p(x, y) must satisfy which condition?

∑_x ∑_y p(x, y) = 1 and p(x, y) ≥ 0 for all x, y (Correct answer)
p(x, y) = p(x) · p(y) for all x, y
∑_x ∑_y p(x, y) = 0
∑_x p(x, y) = 1 for each fixed y

Correct answer: ∑_x ∑_y p(x, y) = 1 and p(x, y) ≥ 0 for all x, y

A valid joint PMF must be non-negative everywhere and sum to 1 over all possible pairs (x, y).

Question 18: A random variable can also be referred to as...

Stochastic (Correct answer)
Clear
Open
Branched

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 19: For independent X ~ Binomial(2, 0.5) and Y ~ Bernoulli(0.5), what is p(X=1, Y=1)?

0.125
0.5
0.375
0.25 (Correct answer)

Correct answer: 0.25

By independence, p(X=1,Y=1) = P(X=1)·P(Y=1) = C(2,1)(0.5)²·0.5 = 0.5·0.5 = 0.25.

Question 20: For a Geometric distribution with p = 0.25, what is P(X = 3)?

(0.75)² × 0.25 ≈ 0.141 (Correct answer)
(0.25)³ ≈ 0.016
0.75 × 0.25 ≈ 0.188
(0.25)² × 0.75 ≈ 0.047

Correct answer: (0.75)² × 0.25 ≈ 0.141

P(X=3) = (1-0.25)^(3-1) × 0.25 = (0.75)² × 0.25 = 0.5625 × 0.25 ≈ 0.141.

Question 21: For the PMF above (P(X=1)=0.3, P(X=2)=0.5, P(X=3)=0.2), what is Var(X)?

3.61
0.7
4.1
0.49 (Correct answer)

Correct answer: 0.49

E[X]=1.9, E[X²]=4.1, Var(X)=4.1-1.9²=4.1-3.61=0.49.

Question 22: The conditional PMF of X given Y = y is defined as:

p(x | y) = p_X(x) · p(x, y)
p(x | y) = p(x, y) · p_Y(y)
p(x | y) = p_X(x) / p(x, y)
p(x | y) = p(x, y) / p_Y(y), provided p_Y(y) > 0 (Correct answer)

Correct answer: p(x | y) = p(x, y) / p_Y(y), provided p_Y(y) > 0

The conditional PMF is defined as p(x | y) = p(x, y) / p_Y(y), analogous to conditional probability P(A|B) = P(A∩B)/P(B).

Question 23: Random variables that can take on any of the countless number of values in an interval are called...

Discrete
Continuous (Correct answer)
Closed
Open

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 24: Which of the following joint PMF tables corresponds to X and Y being independent?

p(1,1)=0.3, p(1,2)=0.1, p(2,1)=0.2, p(2,2)=0.4
p(1,1)=0.06, p(1,2)=0.14, p(2,1)=0.24, p(2,2)=0.56 (Correct answer)
p(1,1)=0.4, p(1,2)=0.1, p(2,1)=0.1, p(2,2)=0.4
p(1,1)=0.25, p(1,2)=0.25, p(2,1)=0.25, p(2,2)=0.25

Correct answer: p(1,1)=0.06, p(1,2)=0.14, p(2,1)=0.24, p(2,2)=0.56

In option B, marginals are p_X(1)=0.2, p_X(2)=0.8, p_Y(1)=0.3, p_Y(2)=0.7, and each cell equals p_X(x)·p_Y(y), confirming independence.

Question 25: If X and Y are independent discrete random variables, what is E[XY]?

E[X²] · E[Y²]
E[X] + E[Y]
E[X] / E[Y]
E[X] · E[Y] (Correct answer)

Correct answer: E[X] · E[Y]

For independent random variables, E[XY] = E[X] · E[Y] because the joint PMF factors into the product of marginals.

Question 26: What is the variance of a Poisson distribution with λ = 7?

49
2.65
7 (Correct answer)
14

Correct answer: 7

For a Poisson distribution, the variance equals λ, so Var(X) = 7.

Question 27: Why can't the Binomial distribution be used when sampling is done without replacement from a small population?

The sample size must equal the population size
The Binomial requires continuous data
Draws are no longer independent, violating the Binomial's independence assumption (Correct answer)
The success probability exceeds 1

Correct answer: Draws are no longer independent, violating the Binomial's independence assumption

Without replacement changes the population composition after each draw, making draws dependent and violating the Binomial's independence requirement.

Question 28: In a Poisson distribution with λ = 3, what is the mean of the distribution?

1.73
3 (Correct answer)
6
9

Correct answer: 3

For a Poisson distribution, both the mean and variance equal λ, so the mean is 3.

Question 29: If P(X = 2) = 0.3, what is P(X ≠ 2)?

0.3
0.7 (Correct answer)
1.3
0

Correct answer: 0.7

P(X ≠ 2) = 1 - P(X = 2) = 1 - 0.3 = 0.7 by the complement rule.

Question 30: Variables which cannot take any value at all are...

Nonrandom (Correct answer)
Calling
Ordinal
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 31: What is the CDF of the Geometric distribution, P(X ≤ k)?

1 - p^k
(1-p)^k
1 - (1-p)^k (Correct answer)
p^k

Correct answer: 1 - (1-p)^k

P(X ≤ k) = 1 - (1-p)^k, representing the probability that the first success occurs by the k-th trial.

Question 32: How does the standard deviation of a Poisson(λ=9) compare to its mean?

The standard deviation is twice the mean
The standard deviation equals the mean
The standard deviation (3) equals the square root of the mean (9) (Correct answer)
The standard deviation is half the mean

Correct answer: The standard deviation (3) equals the square root of the mean (9)

For Poisson, Var(X) = λ = 9, so SD = √9 = 3, which is the square root of the mean.

Question 33: In the Poisson PMF, what role does the constant e (Euler's number ≈ 2.718) play?

It represents the average number of events
It is part of the normalizing factor ensuring probabilities sum to 1 (Correct answer)
It scales the number of trials
It defines the upper bound of the distribution

Correct answer: It is part of the normalizing factor ensuring probabilities sum to 1

The e^-λ term in the Poisson PMF is the normalizing constant that ensures all probabilities sum to exactly 1.

Question 34: Two independent Poisson random variables X ~ Poisson(λ₁) and Y ~ Poisson(λ₂) are added. What is the distribution of X + Y?

Poisson(λ₁ - λ₂)
Poisson(λ₁ + λ₂) (Correct answer)
Normal(λ₁ + λ₂, λ₁ + λ₂)
Poisson(λ₁ × λ₂)

Correct answer: Poisson(λ₁ + λ₂)

The sum of independent Poisson random variables is also Poisson with rate equal to the sum of the individual rates.

Question 35: An example of a continuous variable is...

Acceleration
Power
Force
Distance (Correct answer)

Correct answer: Distance

A continuous variable is one that can take on any value within a certain range. In this case, distance is a continuous variable because it can be measured in any unit of length and can have any value within that unit. Acceleration, force, and power, on the other hand, are not continuous variables as they have specific units and can only take on certain values within those units.

Question 36: For the Negative Binomial (total trials until r-th success), what is the PMF?

P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r) (Correct answer)
P(X=k) = (1-p)^(k-r) × p^r
P(X=k) = C(r+k-1, k) × p^r × (1-p)^k
P(X=k) = C(k,r) × p^r × (1-p)^(k-r)

Correct answer: P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r)

P(X=k) = C(k-1, r-1) × p^r × (1-p)^(k-r) for total trials needed, where the last trial is always the r-th success.

Question 37: For a joint PMF, the conditional PMF p(y | x) satisfies which property for each fixed x?

∑_x ∑_y p(y | x) = 0
∑_y p(y | x) = p_X(x) · p_Y(y)
∑_y p(y | x) = 1 (Correct answer)
∑_y p(y | x) = p_X(x)

Correct answer: ∑_y p(y | x) = 1

For each fixed x, p(y | x) is a valid PMF over y, so it must sum to 1 over all values of y.

Question 38: 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)?

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

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 39: How is the CDF F(x) related to the PMF P(X=k) for discrete random variables?

F(x) = Σ P(X=k) for all k ≤ x (Correct answer)
F(x) = dP(X=x)/dx
F(x) = P(X > x)
F(x) = P(X=x)

Correct answer: F(x) = Σ P(X=k) for all k ≤ x

The CDF is the cumulative sum of the PMF: F(x) = P(X ≤ x) = Σ P(X=k) for all k ≤ x.

Question 40: Which term describes (N-n)/(N-1) in the Hypergeometric variance formula?

Normalizing constant
Continuity correction
Finite population correction factor (Correct answer)
Skewness correction

Correct answer: Finite population correction factor

The factor (N-n)/(N-1) is called the finite population correction factor, adjusting for sampling without replacement.

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%

To Pass

Checking for saved progress...

Exam Rules

You can skip questions and return to them later
Flag questions for review before submitting
No feedback shown until you submit the entire exam
Unanswered questions count as wrong — answer everything
10 pretest questions are mixed in and don't affect your score
Timer auto-submits when time runs out
Your progress is auto-saved every 30 seconds