Questions: Find the expected value of the random variable. x 2 3 4 5 P(x) 0.3 0.4 0.2 0.1 What is the expected value? (Type an integer or a decimal.)

Solution Steps

The expected value (mean) of the random variable \( X \) is calculated using the formula:

\[ E(X) = \sum (x \cdot P(x)) \]

Substituting the values:

\[ E(X) = 2 \times 0.3 + 3 \times 0.4 + 4 \times 0.2 + 5 \times 0.1 \]

Calculating each term:

\[ E(X) = 0.6 + 1.2 + 0.8 + 0.5 = 3.1 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = E(X^2) - (E(X))^2 \]

First, we calculate \( E(X^2) \):

\[ E(X^2) = \sum (x^2 \cdot P(x)) \]

Calculating each term:

\[ E(X^2) = 2^2 \times 0.3 + 3^2 \times 0.4 + 4^2 \times 0.2 + 5^2 \times 0.1 \]

\[ E(X^2) = 4 \times 0.3 + 9 \times 0.4 + 16 \times 0.2 + 25 \times 0.1 \]

\[ E(X^2) = 1.2 + 3.6 + 3.2 + 2.5 = 10.5 \]

Now, substituting back to find the variance:

\[ \sigma^2 = 10.5 - (3.1)^2 = 10.5 - 9.61 = 0.89 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{0.89} \approx 0.943 \]

Final Answer