Statistics
Hypergeometric Distribution — Variance
nta_pyq_2024_apr
Grade 11
Question:
From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the variance of $X$ is $\sigma^2$, then $96\sigma^2$ is equal to _____
Step-by-Step Solution
Key Concept: $X$ follows hypergeometric distribution. $P(X=k)=\dfrac{\binom{3}{k}\binom{7}{5-k}}{\binom{10}{5}}$. Compute $P(X=0)=\frac{7}{15}$, $P(X=1)=\frac{5}{12}$... actually use $\mu=\frac{18}{12}=\frac{3}{2}$ and $\Sigma p_i x_i^2=\frac{34}{12}$.
Variance $\sigma^2=\frac{7}{12}$. $96\sigma^2=56$.
Correct Answer: 56