Question

i. Suppose X ~ B (10,0.99). Use an appropriate procedure involving Poisson random variables to estimate P ({4.2 < X < 7.4}). Also estimate this using the binomial random variable directly. Are the answers close?

Answer 1

If X ~ Bin(10, 0.99)

We have, E(X) = np = 10 * 0.99 = 9.9 < 10.

The mean of Poisson random variable is also 9.9 if we want to approximate the Binomial distribution with Poisson.

So, approximately X ~ Poi(9.9)

The required probability = P(4.2 $\leq$ X $\leq$ 7.4) = P(X = 5) + P(X = 6) + P(X = 7) [As, X is a discrete random variable] =
Le-9.9(9.9)
= P(X $\leq$ 7) - P(X $\leq$ 4) = 0.2294 - 0.0312 = 0.1982.

Using Binomial distribution, the required probability = P(4.2 $\leq$ X $\leq$ 7.4) =
Σ(9) (0.99) (1 – 0.99) 10-
= P(X $\leq$ 7) - P(X $\leq$ 4) = 0.000114 - 0.000001 = 0.000113.

Since, the answers are not close, the approximation is not good in this case.