Define discrete random variables

A discrete random variable assigns a numerical value to each outcome of a random experiment; different outcomes may receive the same value, and its possible values form a finite or countable set. Its probability distribution is represented by a probability mass function, in which probabilities are nonnegative and sum to 1, supporting calculation of event probabilities and measures such as expected value. Continuous random variables and probability density functions are not included.

Detailed Explanation: Define discrete random variables

A discrete random variable is a rule that assigns a numerical value to every outcome of a random experiment. Its possible values are finite or can be listed one at a time.

Worked example: Tossing two coins

Suppose two fair coins are tossed. The outcomes are

\{HH, HT, TH, TT\}.

Define the random variable $X$ as:

$X =$ the number of heads obtained.

Step 1: Assign a value to each outcome

Outcome	Number of heads, $X$
$HH$	$2$
$HT$	$1$
$TH$	$1$
$TT$	$0$

Notice that different outcomes, $HT$ and $TH$ , receive the same value. This is allowed.

Step 2: List the possible values

The possible values of $X$ are

0,\ 1,\ 2.

Therefore, $X$ is a discrete random variable because its possible values can be listed.

Step 3: Create the probability mass function

Each outcome has probability $\frac14$ . Combine the probabilities of outcomes that give the same value:

$x$	$P(X=x)$
$0$	$\frac14$
$1$	$\frac24=\frac12$
$2$	$\frac14$

This table is the probability mass function of $X$ . It tells us the probability of each possible value.

Check that the probabilities are valid:

Each probability is nonnegative.
Their sum is $1$ :

\frac14+\frac12+\frac14=1.

For example, the probability of getting at least one head is

P(X\ge 1)=P(X=1)+P(X=2) =\frac12+\frac14 =\frac34.

Thus, to define a discrete random variable, identify the outcomes, assign a numerical value to each outcome, list the possible values, and give the probability of each value.

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Probability Random Variables - Experiment and X Definition to Outcome Table

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