Statistics
Mean and variance of hypergeometric distribution
nta_pyq_2023_jan
Grade 11
Question:
There are rotten apples mixed accidentally with seven good apples and four apples are drawn one by one without replacement. Let the random variable X denote the number of rotten apples. If $\mu$ and $\sigma^2$ represent mean and variance of X, respectively, then $10(\mu^2 + \sigma^2)$ is equal to
Step-by-Step Solution
Key Concept: The number of rotten apples is not given explicitly — from the solution, the distribution table gives $\mu = \frac{6}{2} = 3$... Wait — re-reading the solution: $\sum xP(x) = \frac{6}{2}$? Actually $\mu = 1$ and $\sigma^2 + \mu^2 = 2$, so $10(\mu^2 + \sigma^2) = 20$. The problem likely has 3 rotten apples among 10 total.
From the probability table: $\mu = \sum xP(x) = 1$, $\sigma^2 + \mu^2 = \sum x^2 P(x) = 2$. So $10(\mu^2 + \sigma^2) = 10 \times 2 = 20$.
Correct Answer: 1