Question

Which of these statements is not true about the variance in a binomial distribution B(n, p)? __________

A. For a fixed p, the variance increases as n increases.

B. For a fixed n, the variance is maximum when p = 0.5.
C. The variance depends only on n.
D. The variance is constant for a specific n and p.
E. None of these are true.

Answer 1

Answer on question 38607 – Math – Statistic and Probability

Which of these statements is not true about the variance in a binomial distribution $B(n, p)$?

A. For a fixed $p$, the variance increases as $n$ increases.

B. For a fixed $n$, the variance is maximum when $p = 0.5$.

C. The variance depends only on $n$.

D. The variance is constant for a specific $n$ and $p$.

E. None of these are true.

Solution.

A: The variance in a binomial distribution can be found by the formula

$$D(Y) = np(1 - p).$$

For a fixed $p$, the variance increases as $n$ increases. A is true.

B: To find the maximum of variance we should find derivative of variance function with respect to $p$.

$$D(Y) = np - np^2$$ $$D'(Y) = n - 2np = n(1 - 2p) = 0$$ $$p = 0.5.$$

Therefore, for a fixed $n$, the variance is maximum when $p = 0.5$. B is true.

C. As we can see from our formula, the variance depends on $n$ and $p$. C is not true.

D: If $n$ and $p$ are constants then the variance is constant. D is true.

Answer: C.