Question

If the p.m.f. of a random variable $X$ is given by
\[ \begin{array}{c|ccc} x & 0 & 1 & 2 \\ P(X=x) & q^2 & 2pq & p^2 \end{array} \] then, the standard deviation of $X$ is (given $p+q=1$)

• A. $2\sqrt{q}$
• B. $\sqrt{2pq}$
• C. $2\sqrt{p}$
• D. $\sqrt{pq}$

Hint:

Always verify that probabilities sum to $1$ before calculating mean and variance.

Answer 1

The Correct Option is B

Step 1: Calculating the mean $E(X)$.
\[ E(X)=0\cdot q^2+1\cdot2pq+2\cdot p^2=2pq+2p^2=2p(p+q)=2p \]
Step 2: Calculating $E(X^2)$.
\[ E(X^2)=0^2\cdot q^2+1^2\cdot2pq+2^2\cdot p^2=2pq+4p^2 \]
Step 3: Finding the variance.
\[ \text{Var}(X)=E(X^2)-[E(X)]^2 \] \[ =(2pq+4p^2)-(2p)^2=2pq \]
Step 4: Finding the standard deviation.
\[ \sigma=\sqrt{\text{Var}(X)}=\sqrt{2pq} \]
Step 5: Conclusion.
The standard deviation of $X$ is $\sqrt{2pq}$.