Given the following probability distribution for the discrete random variable $y$: $\begin{array}{cccc}y& 1& 2& 3\\ P\left(y\right)& 0.2& 0.5& 0.3\end{array}$ What is the expected value of $y$?

• A. $2.5$
• B. $3.0$
• C. $2.1$
• D. $1.8$

Answer: (C)

Verified step by step guidance

1. Identify the values of the discrete random variable \( y \) and their corresponding probabilities from the given distribution. Here, \( y = 1, 2, 3 \) with probabilities \( P(y=1) = 0.2 \), \( P(y=2) = 0.5 \), and \( P(y=3) = 0.3 \).
2. Recall the formula for the expected value (mean) of a discrete random variable: \[ E(Y) = \sum_{i} y_i \times P(y_i) \]
3. Multiply each value of \( y \) by its corresponding probability: \[ 1 \times 0.2, \quad 2 \times 0.5, \quad 3 \times 0.3 \]
4. Sum all the products obtained in the previous step to find the expected value: \[ E(Y) = (1 \times 0.2) + (2 \times 0.5) + (3 \times 0.3) \]
5. The result from the summation is the expected value of \( y \). This value represents the long-run average or mean outcome of the random variable \( y \) based on its probability distribution.