Questions: The following is a probability distribution for a discrete random variable X. X -1 0 5 10 P(X) 0.2 0.5 0.2 0.1 The mean (expected value) of X is -2 1.8 2 -1.8 3.5

The following is a probability distribution for a discrete random variable X.

X -1 0 5 10
P(X) 0.2 0.5 0.2 0.1

The mean (expected value) of X is
-2
1.8
2
-1.8
3.5

Transcript text: The following is a probability distribution for a discrete random variable $X$. \begin{tabular}{|l|l|l|l|l|} \hline$X$ & -1 & 0 & 5 & 10 \\ \hline$P(X)$ & 0.2 & 0.5 & 0.2 & 0.1 \\ \hline \end{tabular} The mean (expected value) of $X$ is -2 1.8 2 -1.8 3.5

Solution

Solution Steps

Step 1: Calculate the Mean (Expected Value)

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

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

Substituting the values:

\[ E(X) = (-1 \times 0.2) + (0 \times 0.5) + (5 \times 0.2) + (10 \times 0.1) = -0.2 + 0 + 1 + 1 = 1.8 \]

Step 2: Calculate the Variance

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

\[ \sigma^2 = \sum ((x_i - E(X))^2 \cdot P(X = x_i)) \]

Substituting the values:

\[ \sigma^2 = (-1 - 1.8)^2 \times 0.2 + (0 - 1.8)^2 \times 0.5 + (5 - 1.8)^2 \times 0.2 + (10 - 1.8)^2 \times 0.1 \]

Calculating each term:

\[ = (-2.8)^2 \times 0.2 + (-1.8)^2 \times 0.5 + (3.2)^2 \times 0.2 + (8.2)^2 \times 0.1 \] \[ = 7.84 \times 0.2 + 3.24 \times 0.5 + 10.24 \times 0.2 + 67.24 \times 0.1 \] \[ = 1.568 + 1.62 + 2.048 + 6.724 = 11.96 \]

Step 3: Calculate the Standard Deviation

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

\[ \sigma = \sqrt{\sigma^2} = \sqrt{11.96} \approx 3.458 \]