FAD1015 Week 4 — Discrete Random Variables (PDF & CDF)
Week 4 covers the foundation of discrete random variables, introducing the probability distribution function (pdf) in Lecture 7 and the cumulative distribution function (CDF) in Lecture 8. Source file: FAD1015 Week 4 discrete pdf cdf.pdf
Summary
This lecture introduces discrete random variables and two ways to characterise their distributions: the probability distribution function $f(x)=P(X=x)$, which gives the probability of each individual outcome, and the cumulative distribution function $F(x)=P(X\leq x)$, which gives the probability of all outcomes up to a given value. The lectures emphasise tabular and functional representations, verification of valid pdfs, and conversion between pdf and CDF.
Key Concepts
- Probability Distributions — Random variable framework
- Discrete Random Variable — countable outcomes with associated probabilities
- Probability Mass Function — $f(x)=P(X=x)$ (lecture calls this the probability distribution function)
- Cumulative Distribution Function — $F(x)=P(X\leq x)$
L7: Probability Distribution Function
1. Random Variable
A variable is a quantity which may take more than one value.
A discrete random variable is a variable which can take individual values each with a given probability. The values are usually the outcome of an experiment.
Examples:
| Experiment | Possible values |
|---|---|
| Score on a fair die | $1,2,3,4,5,6$ |
| Number of heads in 3 coin tosses | $0,1,2,3$ |
| Profit (RM) in a game with entry fee RM 10 and prizes RM 50, RM 100 | $-10, 40, 90$ |
| Number of tosses of a coin until a tail occurs | $1,2,3,\dots$ |
2. Notation
- Random variables are denoted by upper-case letters: $X, Y, R, \dots$
- Particular values are denoted by lower-case letters: $x, y, r, \dots$
- The probability that $X$ takes a particular value $x$ is written $P(X=x)$
- For a fair die: $P(X=4)=\dfrac{1}{6}$
- If the values are $x_1, x_2, \dots, x_n$, the probabilities can be summarised as $p_i$ where $i=1,2,\dots,n$: $$p_1=P(X=x_1),\quad p_2=P(X=x_2),\quad \dots$$
3. Probability Distribution
A probability distribution is a list of all possible values of the discrete random variable $X$, together with their associated probabilities. It can be presented as a table or a function.
Tabular form (fair die):
| $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| $P(X=x)$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ | $\dfrac{1}{6}$ |
Functional form: $$P(X=x)=f(x)=\begin{cases}\dfrac{1}{6}, & x=1,2,3,4,5,6 \[6pt] 0, & \text{otherwise}\end{cases}$$
Here $f(x)$ is called the probability distribution function (pdf).
4. Properties of a Probability Distribution Function
For $f(x)$ to be a valid pdf, it must satisfy:
- Boundedness: $0\leq P(X=x)\leq 1$ for every value of $x$
- Total probability: $\displaystyle\sum_{\text{all }x} P(X=x)=1$
5. Examples
Example 1 — Validating a pdf
Determine whether each function is a probability distribution function.
(a) $f(x)=\dfrac{x^2}{55},\quad x=1,2,3,4,5$
(b) $f(x)=\dfrac{x}{5},\quad x=-1,0,1,2,3$
(c) $f(x)=\dfrac{2x-3}{10},\quad x=2,4,6$
Approach: Check non-negativity for every $x$ and verify that the probabilities sum to 1.
Example 2 — Computing probabilities from a pdf
The random variable $X$ has pdf $P(X=x)=\dfrac{x^2}{14}$ for $x=0,1,2,3$.
Construct the probability distribution table and find:
- (a) $P(X=2)$
- (b) $P(X>1)$
- (c) $P(X\leq 2)$
Distribution table:
| $x$ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| $P(X=x)$ | $0$ | $\dfrac{1}{14}$ | $\dfrac{4}{14}$ | $\dfrac{9}{14}$ |
Check: $0+\dfrac{1}{14}+\dfrac{4}{14}+\dfrac{9}{14}=\dfrac{14}{14}=1$.
Example 3 — Finding a missing constant
Aleeya plays with a biased five-sided spinner marked $1,2,3,4,5$. The pdf of her score $X$ is:
| $x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $P(X=x)$ | $0.15$ | $0.24$ | $k$ | $0.25$ | $0.19$ |
Find:
- (a) $k$
- (b) $P(X\geq 4)$
- (c) $P(X<5)$
- (d) $P(2<X\leq 4)$
Solution for (a): $0.15+0.24+k+0.25+0.19=1\implies k=0.17$.
Example 4 — Hypergeometric-type setting
A basket contains 12 peppers: 3 red, 4 green, 5 yellow. Three peppers are taken at random without replacement.
- (a) Find the probability that the three peppers are all different colours.
- (b) Show that the probability exactly 2 peppers are green is $\dfrac{12}{55}$.
- (c) Let $X$ be the number of green peppers taken. Draw up the probability distribution table for $X$.
Values of $X$: $0,1,2,3$.
Example 5 — Piecewise pdf with unknown constant
The pdf of $X$ is $$f(x)=\begin{cases}kx^2, & x=0,1,2,3 \ k, & x=4,5\end{cases}$$
Find:
- (a) the value of $k$
- (b) $P(0<X\leq 3)$
- (c) $P(|X-2|<1)$
- (d) $P(X\text{ is even})$
Approach for (a): Sum all probabilities and set equal to 1: $$k(0^2+1^2+2^2+3^2)+k+k = k(0+1+4+9)+2k = 15k+2k = 17k = 1 \implies k=\frac{1}{17}$$
L8: Cumulative Distribution Function
1. Definition
For a discrete random variable $X$ with pdf $P(X=x)$ where $x=x_1,x_2,\dots,x_n$, the cumulative distribution function $F(t)$ is defined as
$$F(t)=P(X\leq t)=\sum_{x_1}^{t} P(X=x)$$
Fair die example:
| $x$ | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| $F(x)$ | $\dfrac{1}{6}$ | $\dfrac{2}{6}$ | $\dfrac{3}{6}$ | $\dfrac{4}{6}$ | $\dfrac{5}{6}$ | $1$ |
Written as a piecewise function: $$F(x)=\begin{cases}0, & x<1 \ \dfrac{1}{6}, & 1\leq x<2 \ \dfrac{2}{6}, & 2\leq x<3 \ \dfrac{3}{6}, & 3\leq x<4 \ \dfrac{4}{6}, & 4\leq x<5 \ \dfrac{5}{6}, & 5\leq x<6 \ 1, & x\geq 6\end{cases}$$
2. Key Relationships
For any discrete random variable with CDF $F$:
$$P(a<X\leq b)=P(X\leq b)-P(X\leq a)=F(b)-F(a)$$
$$P(X=b)=F(b)-F(a)$$
(Lecture notation: the second formula uses $a$ to denote the value immediately preceding $b$ in the support of $X$.)
3. Examples
Example 6 — CDF from a table (spinner revisited)
Using the spinner from Example 3 with $k=0.17$:
| $x$ | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| $P(X=x)$ | $0.15$ | $0.24$ | $0.17$ | $0.25$ | $0.19$ |
- (b) Find the cumulative distribution of $X$.
- (c) Find:
- i. $P(X<3)$
- ii. $P(2<X\leq 5)$
- iii. $P(2\leq X<5)$
Example 7 — Recovering the pdf from the CDF
$X$ takes values $0,1,2,3$ with CDF: $$F(x)=\begin{cases}0, & x<0 \ 0.2, & 0\leq x<1 \ 0.6, & 1\leq x<2 \ 0.9, & 2\leq x<3 \ 1, & x\geq 3\end{cases}$$
- (a) Find:
- i. $P(X\leq 1)$
- ii. $P(X>1)$
- iii. $P(X=1)$
- iv. $P(0<X\leq 2)$
- v. $P(1<X<3)$
- (b) Find the pdf $f(x)$.
Key idea: $P(X=x_i)=F(x_i)-F(x_{i-1})$.
Example 8 — Car sales
A car dealer records the number of cars sold per day. The relative frequencies give:
| Number of cars, $x$ | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| $P(X=x)$ | $0.1$ | $0.1$ | $0.2$ | $0.3$ | $0.2$ | $0.1$ |
- (a) Find the probability that 2 cars or less will be sold the next day.
- (b) Find the cumulative distribution function.
Example 9 — Peppers (full pdf and CDF)
Same setting as Example 4 (12 peppers: 3 red, 4 green, 5 yellow; 3 drawn without replacement).
- (a) Let $X$ be the number of green peppers taken. Draw up the probability distribution table for $X$.
- (b) Find the cumulative frequency distribution of green peppers taken.
Related Topics
- FAD1015 Week 3 — Independent Events & Bayes' Theorem — prerequisite probability
- FAD1015 Week 5 — Mean & Variance (Discrete & Continuous) — extends to moments
- FAD1015 Tutorial 1-6 — Counting & Probability Fundamentals — practice problems
Related Course Page
- FAD1015 - Mathematics III