Suppse that the number of justices appointed in a year is a

Suppse that the number of justices appointed in a year is a random variable with mean mu=0.5223 and variance sigma^2=0.5556.

A. Let the random variable X be the number of justices appointed in a year. Suppose that X has a binomial distribution with n=4. Find p (probability) to 4 decimals using mu.

B. Let the random variable Y be the number of justices appointed in a year. Suppose that Y has a Poisson distribution. Find lamda to 4 decimals using mu.

Solution

m = n * p = 0.5223

s^2 = n*p*(1-p) = 0.5556

So,

taking values from each other and solving,

0.5223*(1-p) = 0.5556

1-p = 1.064

p = 0.064

q = 0.936

n = 8

P(x=4) = 8C4 * 0.064^4 * 0.936^4 = 0.0009014

2.

m = 0.5223

poission eq.

P(x=r) = e^(-m) * m^(r) / r!

P(Y) = e^(-0.5223) * 0.5223^y / y!

taking y=4

P(4) = e^(-0.5223) * (0.5223)^4/4!

P(x=4) = 0.001839