Questions: PMFs, PDFs, and CDFs
Before attempting these questions it is highly recommended that you read Guide: PMFs, PDFs, and CDFs.
Q1
For each of the scenarios below, determine if the given distribution is a valid PMF and answer the following questions.
1.1.
Let \(X\) be the random variable representing the result of rolling a biased four sided-die. The PMF of \(X\) is given by:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| \(P(X=x)\) | \(\dfrac{1}{10}\) | \(\dfrac{1}{5}\) | \(\dfrac{1}{2}\) | \(\dfrac{1}{5}\) |
What is \(P(X = 4)\)?
1.2.
A discrete random variable \(X\) has five possible outcomes (\(1, 2, 3, 4,\) or \(5\)), and the PMF is given by:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) |
|---|---|---|---|---|---|
| \(P(X=x)\) | \(0.25\) | \(0.35\) | \(0.05\) | \(0.2\) | \(0.1\) |
What is the probability of \(X = 3\) or \(X = 4\)?
1.3.
A coin is tossed, where the probability of tails is \(70%\) and heads is \(30%\). Let \(X\) represent the result of the coin toss. Complete the table below:
| \(x\) | Heads | Tails |
|---|---|---|
| \(P(X=x)\) |
1.4.
A discrete random variable \(X\) has the possible outcomes \(1, 2, 3, 4, 5, 6,\) or \(7\), with the following PMF:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) |
|---|---|---|---|---|---|---|---|
| \(P(X=x)\) | \(0.1\) | \(0.05\) | \(0.05\) | \(0.3\) | \(0.25\) | \(0.75\) | \(0.35\) |
Is this a valid PMF? Justify your answer either way.
1.5.
A bag contains \(5\) red, \(3\) blue, and \(2\) green sweets from a sweet shop. Let \(X\) represent the color of a randomly picked sweet:
What is the probability of picking a blue sweet?
Construct the PMF for this scenario by completing the table:
| \(x\) | Red | Blue | Green |
|---|---|---|---|
| \(P(X=x)\) |
1.6.
The PMF for a random variable \(X\) is given as:
| \(x\) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| \(P(X=x)\) | \(p\) | \(2p\) | \(3p\) | \(4p\) |
For what value of \(p\) is this a valid PMF?
For this value of \(p\), what is \(P(X = 3)\)?
Q2
For each of the scenarios below, determine if the given distribution is a valid PDF and answer the following questions.
2.1.
Let \(X\) be a continuous random variable on the interval \([0, 2]\) with the PDF:\[f(x) =\begin{cases} \dfrac{1}{2} & \textsf{if } 0 \leq x \leq 2 \\[0.5em] 0 & \textsf{otherwise} \end{cases}\]
What is the probability that \(X\) lies between 1 and 2?
2.2.
Let \(X\) be a continuous random variable with the PDF: \[f(x) = \begin{cases} 2x & \textsf{if } 0 \leq x \leq 1, \\[0.5em] 0 & \textsf{otherwise}.\end{cases}\]
What is the probability that \(X\) lies between \(0.5\) and \(1\)?
What is \(P(0.25 \leq X \leq 0.75)\)?
2.3.
Let \(X\) be a continuous random variable uniformly distributed between \(3\) and \(7\). The PDF is: \[f(x) =\begin{cases}\dfrac{1}{4} & \textsf{if } 3 \leq x \leq 7 \\[0.5em] 0 & \textsf{otherwise}\end{cases}\]
What is the probability that \(X\) lies between \(3\) and \(6\)?
2.4.
The PDF of a random variable \(X\) is given by: \[f(x) =\begin{cases}\dfrac{1}{9} & \textsf{if } 1 \leq x \leq 4 \\[0.5em]\dfrac{1}{4} & \textsf{if } 5 \leq x \leq 7 \\[0.5em] 0 & \textsf{otherwise}\end{cases}\]
Is this a valid PDF? Justify your answer either way.
2.5.
Consider the PDF:
\[ f(x) = \begin{cases} k x^2 & \textsf{if } 0 \leq x \leq 1 \\[0.5em] 0 & \textsf{otherwise} \end{cases} \]
For what value of \(k\) is this a valid PDF?
For this value of \(k\), what is \(P(0.2 \leq x \leq 0.3)\)?
2.6.
The PDF of \(X\) is given by:
\[ f(x) = \begin{cases} 4x & \textsf{if } 0 \leq x < 0.5, \\[0.5em] 4 - 4x & \textsf{if } 0.5 \leq x < 0.75, \\[0.5em] 0.5 & \textsf{if } 0.75 \leq x \leq 1, \\[0.5em] 0 & \textsf{otherwise}. \end{cases} \]
Is this a valid PDF? Justify your answer either way.
Q3
For each of the scenarios below, answer the following questions.
3.1.
In a scenario involving a discrete random variable, the following CDF is given:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) |
|---|---|---|---|---|
| \(P(X=x)\) | \(0.1\) | \(0.3\) | \(0.5\) | \(1\) |
What is \(F(3)\)?
What is \(P(X > 2)\)?
3.2.
For the random variable uniformly distributed on \([0, 2]\) as seen in Q2.2:
Calculate the CDF at values \(0.5\), \(1\), and \(2\).
What is \(F(3)\)?
3.3.
For the PDF given in Q2.3:
Calculate the CDF at points \(x=4\), \(x=5\), and \(x=6\).
What is \(P(X > 5)\)?
3.4.
The CDF of \(X\) for a scenario is given by:
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
|---|---|---|---|---|---|---|
| \(P(X=x)\) | \(0.1\) | \(0.2\) | \(0.5\) | \(0.4\) | \(0.8\) | \(1\) |
Is this a valid CDF? Justify your answer either way.
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 12/24 by Sophie Chowgule as part of a University of St Andrews VIP project.