91Ó°ÊÓ

In probability and statistics, a discrete random variable is a type of random variable that can take on a finite or countably infinite set of values. Unlike continuous random variables, which can take any value in a given range, discrete random variables have distinct, separate values. Examples of discrete random variables include the roll of a dice, number of students in a class, or even the number of cars passing a traffic signal in a day. To describe a discrete random variable, we need to know:

• The possible values it can take.
• The probabilities associated with each of these values.

In our exercise, the random variable X takes the values 0, 1, and 2, with respective probabilities 0.2, 0.3, and 0.5. This means X is discrete because it has distinct possible outcomes and associated probabilities. Understanding this helps us apply the expectation formula correctly.

The expectation formula, also known as the expected value or mean of a random variable, gives us a measure of the central tendency of that random variable. For a discrete random variable, it's calculated using:
\[ E(X) = \sum_x x \cdot P(X=x) \] This formula might look complicated, but it's quite straightforward. It simply sums up the products of each possible value the variable can take (like 0, 1, or 2 in our example) and its respective probability. Here's a step-by-step of how we use this formula in our solution:

• Step 1: List out all values X can take.
• Step 2: Identify the probability for each value.
• Step 3: Multiply each value by its probability.
• Step 4: Add all those products together to get the expectation.

So, for our exercise, it becomes: \[ E(X) = (0 \cdot 0.2) + (1 \cdot 0.3) + (2 \cdot 0.5) = 0 + 0.3 + 1.0 = 1.3 \]

A probability distribution describes how the probabilities are distributed over the values of a random variable. For a discrete random variable, its probability distribution specifies each possible value the variable can take and the probability that it takes each value.
It is often presented in a list or a formula format, and in our exercise, it looks like this:
\[ \begin{aligned} P(X=0) &= 0.2 \ P(X=1) &= 0.3 \ P(X=2) &= 0.5 \end{aligned} \] Understanding the probability distribution is crucial because:

• It informs us of all possible outcomes for the variable.
• It gives us their respective probabilities, which we need for calculations like the expectation.

In our example, knowing the distribution allows us to calculate the expectation accurately by plugging in the values and probabilities into the expectation formula. Probability distributions provide the framework that supports all such calculations in probability theory.