FAD1015: Mathematics III — Tutorial 5
Centre for Foundation Studies in Science
Universiti Malaya
Session 2025/2026
Topic: Discrete Random Variable
Probability Distribution Function (pdf)
Question 1
Determine and explain if the following functions are probability functions or otherwise?
(a) $P(X = x) = \frac{2x-1}{16}$, $x = 1, 2, 3, 4$
(b) $P(X = x) = \frac{x}{15}$, $x = -1, 0, 1, 2$
(c) $P(X = x) = \frac{2}{3(1+2^x)}$, $x = 0, 1, 2$
Question 2
$X$ is a random variable with probability distribution:
$$P(X = x) = \begin{cases} 0.1 & \text{for } x = 2, 4, 6, 8 \ 0.2 & \text{for } x = 1, 3, 5 \ 0 & \text{otherwise} \end{cases}$$
Show that $X$ is a discrete random variable. Hence, find:
(a) $P(X = 4)$
(b) $P(X < 4)$
(c) $P(3 \leq X < 6)$
Question 3
A discrete random variable has probability function:
$$P(X = x) = \begin{cases} 0.1 & \text{for } x = -1, 0, 5 \ a & \text{for } x = 1, 3 \ 0.3 & \text{for } x = 4 \end{cases}$$
(a) Write out the probability distribution table in terms of $a$.
(b) Find $a$.
(c) Find $P(X \geq 3)$.
(d) Find the Cumulative Distribution Function of $X$.
Question 4
A drawer contains 8 white socks and 4 yellow socks. Alisha takes two socks at random from the drawer one after the other.
(a) Show that the probability that Alisha takes one white sock and one yellow sock is $\frac{16}{33}$.
(b) The discrete random variable $Y$ is the number of white socks taken. Find the probability distribution and cumulative distribution function of $Y$.
Question 5
The discrete random variable X has the probability distribution shown in the table.
| $X = x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $P(X = x)$ | 0.2 | 0.25 | 0.4 | $k$ | 0.05 |
(a) Find the value of $k$.
(b) Find:
i. $P(1 \leq X \leq 3)$
ii. the probability that X is at least 3.
iii. $P(2 < X < 5)$
(c) Find the cumulative distribution function of $X$.
Question 6
Two fair tetrahedral dice each have faces marked 1, 2, 3 and 4. The two dice are thrown together. The random variable $D$ is zero if both dice land on the same number. If the dice do not land on the same number, then $D$ is the positive difference between the numbers on which they land.
(a) Complete the table below.
| Dice 2 \ Dice 1 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 1 | ||||
| 2 | 0 | |||
| 3 | ||||
| 4 |
(b) Find:
i. $P(D \leq 1)$
ii. $P(D > 2)$
(c) Find the probability distribution and cumulative distribution function of $D$.
Question 7
The discrete random variable X has the cumulative distribution function shown in the table.
$$F(x) = \begin{cases} 0, & x < 0.1 \ 0.05, & 0.1 \leq x < 0.2 \ 0.3, & 0.2 \leq x < 0.3 \ 0.6, & 0.3 \leq x < 0.4 \ 0.75, & 0.4 \leq x < 0.5 \ 1, & x \geq 0.5 \end{cases}$$
(a) Find:
i. $P(X \leq 0.4)$
ii. $P(X > 0.3)$
iii. $P(X = 0.3)$
(b) Find the probability distribution function of $X$.
Related Concepts
- Discrete Random Variable — random variable with countable outcomes
- Probability Mass Function — pdf for discrete random variables
- Cumulative Distribution Function — probability that X takes a value less than or equal to x
- Probability Distribution — function giving probabilities of different outcomes
Source: FAD1015 Questions T1-T6 _20252026.pdf