Probability
Random Variable — Mean, Variance and a/b
nta_pyq_2026_jan
Grade 12
Question:
A random variable $X$ takes values $0,1,2,3$ with probabilities $\dfrac{2a+1}{30}$, $\dfrac{8a-1}{30}$, $\dfrac{4a+1}{30}$, $b$ respectively, where $a,b\in\mathbb{R}$. Let $\mu$ and $\sigma$ respectively be the mean and standard deviation of $X$ such that $\sigma^2+\mu^2=2$. Then $\dfrac{a}{b}$ is equal to:
Step-by-Step Solution
Key Concept: Sum of probabilities $=1$: $b=\tfrac{29-14a}{30}$. $\sigma^2+\mu^2=E(X^2)=2$. Compute $E(X^2)=\tfrac{-102a+264}{30}=2\Rightarrow a=2$.
$a=2$, $b=\tfrac{1}{30}$. $\tfrac{a}{b}=60$.
Correct Answer: 3