Porisma

Worked solution

Infer p from variance and compare with the mean

The random variable X∼B(16,p)X\sim B(16,p) where p<0.5p<0.5. If the variance of XX is 3.36, find the value of pp. Find also the probability that XX is less than the mean.

CJC · Discussion · Question 5

Answer

p=0.3p=0.3; P(X<E[X])=0.450P(X<\mathrm E[X])=0.450 (3 s.f.).

Find p

Use Var⁡(X)=16p(1−p)=3.36\operatorname{Var}(X)=16p(1-p)=3.36. Rearrange to 16p2−16p+3.36=016p^2-16p+3.36=0. The quadratic formula gives p=[16±256−4(16)(3.36)]/32p=[16\pm\sqrt{256-4(16)(3.36)}]/32, so p=0.3 or 0.7. Reject 0.7 because p<0.5. Hence X∼B(16,0.3)X\sim B(16,0.3).

How did this part go?

Less than the mean

The mean is 16(0.3)=4.816(0.3)=4.8. Since X takes integer values, P(X<4.8)=P(X≤4)=0.4499041198…P(X<4.8)=P(X\leq4)=0.4499041198\ldots, giving 0.450 (3 s.f.).

How did this part go?

Common mistakes

“Below the mean” does not mean X<4 when the mean is 4.8; it includes X=4.

Read the hint