Question

The following table gives the probability distribution of a random variable \(X\): \[ \begin{array}{|c|c|c|c|c|} \hline x_i & 0 & 1 & 2 & 3 \\ \hline P(X=x_i) & 0.1 & 0.3 & 0.5 & 0.1 \\ \hline \end{array} \] The standard deviation of \(X\) is:

• A. \(0.4\)
• B. \(0.42\)
• C. \(0.8\)
• D. \(0.64\)

Hint:

For discrete probability distributions: \[ E(X)=\sum x_i p_i \] and \[ \sigma=\sqrt{E(X^2)-[E(X)]^2} \] Always calculate the mean first before computing variance and standard deviation.

Answer 1

The Correct Option is C

Step 1: Recall the formulas.
For a discrete variable, mean $E(X)=\sum x_iP_i$, variance $=E(X^2)-[E(X)]^2$, and standard deviation is the square root of the variance.
Step 2: Find the mean.
$E(X)=0(0.1)+1(0.3)+2(0.5)+3(0.1)=0+0.3+1+0.3=1.6$.
Step 3: Find $E(X^2)$.
$E(X^2)=0(0.1)+1(0.3)+4(0.5)+9(0.1)=0+0.3+2+0.9=3.2$.
Step 4: Compute the variance.
Variance $=3.2-(1.6)^2=3.2-2.56=0.64$.
Step 5: Take the square root.
Standard deviation $=\sqrt{0.64}=0.8$.
Step 6: State the answer.
So the standard deviation of $X$ is $0.8$. \[ \boxed{0.8} \]