Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > Mathematics > Advanced Math

Probability Distribution Quiz: Spot All Valid Distributions!

Ready for your probability unit test? Identify valid probability distributions now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art banner for probability distribution quiz with charts, graphs, question marks on a sky blue background.

Curious which of the following are valid probability distributions? Test your mastery in this free, fast-paced scored, interactive quiz and identify all valid probability distributions examples while tackling key probability distribution questions. You'll face diverse scenarios inspired by real data and exam-style challenges, making it ideal for anyone preparing for a probability unit test or math enthusiast seeking extra practice. If you've already tried our normal distribution quiz or the binomial distribution test, now's your chance to level up. Dive in, challenge yourself, and prove you've got what it takes!

Which of the following defines a valid probability mass function (pmf) for k = 1, 2, 3, ...?
p(k) = 1/2^k for k ? 1
p(k) = 1/k for k ? 1
p(k) = 1/10^k for k ? 0
p(k) = (-1)^k / 2^k for k ? 1
A valid pmf must assign nonnegative probabilities that sum to 1. The series sum_{k=1 to ?} 1/2^k equals 1 by the geometric series formula. The other options either diverge, include negative values, or sum to a value different from 1.
Which of the following is a valid pmf on the set {1, 2, 3, 4}?
p(k) = k/10 for k = 1, 2, 3, 4
p(k) = 1/k for k = 1, 2, 3, 4
p(k) = 1/3 for k = 1, 2, 3
p(k) = 1/16 for k = 1, 2, 3, 4
A pmf must sum to 1 and be nonnegative. Here, (1 + 2 + 3 + 4) / 10 equals 1, so p(k) = k/10 is valid. The other choices either sum to a value not equal to 1 or assign probabilities outside [0,1].
Which of the following gives a valid pmf on {0, 1, 2}?
p(0)=0.5, p(1)=0.3, p(2)=0.2
p(0)=0.6, p(1)=0.2, p(2)=0.3
p(0)=0.3, p(1)=0.3, p(2)=0.3
p(0)=0.5, p(1)=-0.1, p(2)=0.6
Valid pmfs assign nonnegative probabilities summing to 1. The first option sums exactly to 1 with all values ? 0. The others either exceed 1, sum to less than 1, or include a negative probability.
On the domain {1, 2, 3, 4}, which of these is a valid discrete uniform distribution?
p(k) = 1/4 for k = 1, 2, 3, 4
p(k) = 1/k for k = 1, 2, 3, 4
p(k) = k/8 for k = 1, 2, 3, 4
p(k) = 1/16 for k = 1, 2, 3, 4
A discrete uniform distribution on four outcomes assigns probability 1/4 to each. It is nonnegative and sums to 1. The other options either sum to more or less than 1 or vary by k.
Which of the following is a valid probability density function (pdf) on the interval [0, 1]?
f(x) = 1 for 0 ? x ? 1
f(x) = 2 for 0 ? x ? 1
f(x) = 1 for 0 ? x ? 2
f(x) = 0.5 for 0 ? x ? 1
A pdf must be nonnegative and integrate to 1 over its support. ??¹ 1 dx = 1, so f(x)=1 is valid. The other functions either integrate to a value other than 1 or lie outside the specified interval.
Which of the following describes a valid exponential pdf with rate parameter ? = 1?
f(x) = e^{-x} for x ? 0
f(x) = e^{-2x} for x ? 0
f(x) = 2 e^{-x} for x ? 0
f(x) = 1 - e^{-x} for x ? 0
The exponential pdf with rate 1 is f(x)=e^{-x} for x?0, which integrates to 1. Other forms either integrate to a different constant or represent a CDF instead of a pdf.
Which function defines a valid uniform pdf on [0, 5]?
f(x) = 1/5 for 0 ? x ? 5
f(x) = 1/4 for 0 ? x ? 5
f(x) = 1/5 for 0 ? x ? 4
f(x) = 5 for 0 ? x ? 1
A uniform pdf on [0,5] must equal 1/(5?0)=1/5 across the interval and zero elsewhere. The other choices yield integrals not equal to 1 or have incorrect support.
For the pmf p(k) = c / k² for k = 1, 2, 3, ... , what is the value of c that makes it a valid distribution?
6 / ?²
1 / ?²
3 / ?²
12 / ?²
The normalization constant c must satisfy ????^? 1/k² = ?²/6, so c = 1 / (?²/6) = 6/?². Other choices miscalculate the Riemann zeta value.
Find the constant A so that f(x) = A x for 0 ? x ? 2 (and 0 otherwise) is a valid pdf.
1/2
1/4
1
2
We require ??² A x dx = A·(2²/2) = 2A = 1, so A = 1/2. The other values do not satisfy the integral condition.
Which of these describes a valid mixed distribution combining a point mass at 0 and a continuous part on (0,?)?
P(X=0)=0.2; f(x)=0.8 e^{-x} for x>0
P(X=0)=0.5; f(x)=0.6 e^{-x} for x>0
P(X=0)=0.1; f(x)=0.8 e^{-x} for x>0
P(X=0)=0.2; f(x)=1 e^{-x} for x>0
A mixed distribution must allocate total probability 1. Here 0.2 + 0.8·??^? e^{-x}dx = 0.2 + 0.8·1 = 1. The others either exceed or fall short of 1.
Determine the constant k so that f(x) = k x^{-1/2} for 0 < x ? 1 (and 0 otherwise) is a valid pdf.
1/2
1
2
?2
Normalization requires ??¹ k x^{-1/2} dx = k·2 = 1, so k = 1/2. The other options do not satisfy the integral condition.
0
{"name":"Which of the following defines a valid probability mass function (pmf) for k = 1, 2, 3, ...?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"Which of the following defines a valid probability mass function (pmf) for k = 1, 2, 3, ...?, Which of the following is a valid pmf on the set {1, 2, 3, 4}?, Which of the following gives a valid pmf on {0, 1, 2}?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Identify Valid Probability Distributions -

    Analyze probability distribution questions to determine which examples satisfy the axioms of probability, ensuring probabilities are non-negative and sum to one.

  2. Construct Distribution Tables -

    Create clear probability distribution tables for discrete random variables, organizing outcomes and their probabilities effectively.

  3. Calculate Expected Values -

    Apply formulas to compute the expected value of a random variable from a given distribution table.

  4. Evaluate Invalid Distributions -

    Detect and explain errors in distributions that violate probability rules, such as probabilities outside the 0 to 1 range or sums not equal to one.

  5. Apply Quiz Strategies -

    Use proven techniques to approach probability distributions quiz questions confidently and accurately.

  6. Reinforce Probability Concepts -

    Strengthen understanding of key terms and concepts from your probability unit test, improving overall mastery and retention.

Cheat Sheet

  1. Non-negativity and Total Probability -

    Every probability value must be ≥ 0 and the sum (for discrete) or integral (for continuous) of all outcomes equals 1. A handy mnemonic is "No Negative Vikings" to remember probabilities can't go below zero and must add up to one. This fundamental rule is emphasized in sources like MIT OpenCourseWare and Khan Academy.

  2. Discrete Probability Mass Function (PMF) -

    A discrete PMF assigns a probability p(x) to each outcome x, and ∑x p(x)=1 across its support. When you see a table of probabilities, just add each entry to confirm it hits exactly 1 for a valid probability distribution. Universities like Stanford and UC Berkeley often use simple dice or coin”toss examples to illustrate this.

  3. Continuous Probability Density Function (PDF) -

    In a continuous setting, a PDF f(x) must satisfy f(x)≥0 for all x and ∫_−∞^∞ f(x) dx=1. For example, the normal distribution's bell curve is valid because its total area under the curve is precisely 1. This integral check is covered in depth by sources such as Coursera's calculus and statistics courses.

  4. Expected Value and Variance Formulas -

    The expected value E(X) is given by ∑x x p(x) (discrete) or ∫x f(x) dx (continuous), and variance is Var(X)=E(X²)−[E(X)]². These formulas help you confirm distribution properties and are essential for scoring high on any probability distributions quiz. Check reputable texts like Ross's "Introduction to Probability Models" for worked examples.

  5. Common Valid Probability Distributions Examples -

    Familiarize yourself with classic valid probability distributions examples like the Binomial(n,p), Poisson(λ), and Geometric(p). Always verify that parameters lie in the correct range (e.g., 0≤p≤1) and that their PMFs or PDFs integrate or sum to one. Official documentation from the American Statistical Association provides clear parameter checks for each.

Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Advanced Math Quizzes

Powered by: Quiz Maker