A probability distribution for a discrete random variable $X$ is a list of each possible value for $X$ with the probability that $X$ will have that value when the experiment is run. The likelihood for the probability distribution of a discrete random variable is denoted by $P(X=x)$. The probability distribution of a discrete random variable is also called a discrete probability distribution.
A discrete probability distribution is a mathematical function that assigns probabilities to each possible value of a discrete random variable.
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Example of Probability Distribution of a Discrete Random Variable
Let $X$ be a random variable representing the number of trials obtained when a coin is flipped three times in an experiment. The sample space of the experiment is:
$$HHH, HHT, HTH, THH, HTT, TTH, THT, TTT$$
where $T$ represents the occurrence of Tail and $H$ represents the occurrence of Head in the above experiment.
Then $X$ has 4 possible values: $0, 1, 2, 3$ for the occurrence of head or tail. The probability distribution for $X$ is given as below:
| $X$ | $P(X)$ |
|---|---|
| 0 | $\frac{1}{8}$ |
| 1 | $\frac{3}{8}$ |
| 2 | $\frac{3}{8}$ |
| 3 | $\frac{1}{8}$ |
| Total | $1.0$ |
In a statistics class of 25 students are given a 5-point quiz. 3 students scored 0; 1 student scored 1, 4 students scored 2, 8 students scored 3, 6 students scored 4, and 3 students scored 5. If a student is chosen at random, and the random variable $S$ is the student’s Quiz Score then the discrete probability distribution of $S$ is
| $S$ | $P(S)$ |
|---|---|
| 0 | 0.12 |
| 1 | 0.04 |
| 2 | 0.16 |
| 3 | 0.32 |
| 4 | 0.24 |
| 5 | 0.12 |
| Total | 1.0 |
Note that for any discrete random variable $X$, $0\le P(X) \le 1$ and $\Sigma P(X) =1$.
Finding Probabilities from a Discrete Probability Distribution
Since a random variable can only take one value at a time, the events of a variable assuming two different values are always mutually exclusive. The probability of the variable taking on any number of different values can thus be found by simply adding the appropriate probabilities.
Mean or Expected Value of a Discrete Random Variable
The mean or expected value of a random variable $X$ is the average value that one should expect for $X$ over many trials of the experiment in the long run. The general notation of the mean or expected value of a random variable $X$ is represented as
$$\mu_x\quad \text{ or } E[X]$$
The mean of a discrete random variable is computed using the formula
$$E[X]=\mu_x = \Sigma x\cdot P(X)$$
Example 1
From the above experiment of three Coins the Expected value of the random variable $X$ is
| $X$ | $P(X)$ | $x.P(X)$ |
|---|---|---|
| 0 | $\frac{1}{8}$ | $0 \times \frac{1}{8} = 0$ |
| 1 | $\frac{3}{8}$ | $1 \times \frac{3}{8} = \frac{3}{8}$ |
| 2 | $\frac{3}{8}$ | $2 \times \frac{3}{8} = \frac{6}{8}$ |
| 3 | $\frac{1}{8}$ | $3 \times \frac{1}{8} = \frac{3}{8}$ |
| Total | $1.0$ | $\frac{3}{2} = 1.5$ |
Thus if three coins are flipped a large number of times, one should expect the average number of trials (per 3 flips) to be about 1.5.
Example 2
Similarly, the mean of the random variable $S$ from the above example is
| $S$ | $P(S)$ | $S\cdot P(S)$ |
|---|---|---|
| 0 | 0.12 | $0 \times 0.12 = 0$ |
| 1 | 0.04 | $1 \times 0.04 = 0.04$ |
| 2 | 0.16 | $2 \times 0.16 = 0.32$ |
| 3 | 0.32 | $3 \times 0.32 = 0.96$ |
| 4 | 0.24 | $4\times 0.24 = 0.96$ |
| 5 | 0.12 | $5 \times 0.12 = 0.60$ |
| Total | $1.0$ | $2.88$ |
Note that $2.88$ is the class average on the statistics quiz as well.
Variance and Standard Deviation of a Random Variable
One may be interested to find how much the values of a random variable differ from trial to trial. To measure this, one can define the variance and standard deviation for a random variable $X$. The variance of $X$ random variable is denoted by $\sigma^2_x$ while the standard deviation of the random variable $X$ is just the square root of $\sigma^2_x$. The formulas of variance and standard deviation of a random variable $X$ are:
\begin{align*}
\sigma^2_x &= \Sigma (x – \mu)^2 P(X)\\
\sigma_x &= \sqrt{\Sigma (x – \mu)^2 P(X)}
\end{align*}
Note that the standard deviation estimates the average difference between a value of $x$ and the expected value.
Calculating the Variance and Standard Deviation
The calculation of standard deviation for a random variable is similar to the calculation of weighted standard deviation in a frequency table. The $P(x)$ can be thought of as the relative frequency of $x$. The computation of variance and standard deviation of a random variable $X$ can be made using the following steps:
- Compute $\mu_X$ (mean of the random variable)
- Subtract the mean/average from each of the possible values of $X$. These values are called the deviations of the $X$ values.
- Square each of the deviations calculated in the previous step.
- Multiply each squared deviation (calculated in step 3) by the corresponding probability $P(x)$.
- Sum the results of step 4. The variance of the random variable will be obtained representing $\sigma^2_X$.
- Take the square root of the $\sigma^2_X$ computed in Step 5.
Importance of Discrete Probability Distributions
- Modeling Real-World Phenomena: Discrete Distributions help us understand and model random events in various fields of life such as engineering, finance, and the sciences.
- Decision Making: These distributions provide a framework for making informed decisions under uncertainty.
- Statistical Inference: These are used to make inferences about populations based on sample data.
FAQs about the Probability Distribution of a Discrete Random Variable
- Define the probability distribution.
- What is a random variable?
- What is meant by an expected value or a random variable?
- What is meant by the variance and standard deviation of a random variable?
