Question

outside the United States? Poisson distribution with Xt equal to 3: b. P(r > 3) d. Find the smallest x' so that P(x s x') > 0.50. 5-56. Determine the following values associated with a

Answer 1

We are given the distribution here as:

$\LARGE X \sim Poisson(3)$

$\LARGE P(X = 0) = e^{-3} = 0.0498$

$\LARGE P(X = 1) = 3e^{-3} = 0.1494$

$\LARGE P(X = 2) = \frac{3^2}{2}e^{-3} = 0.2240$

$\LARGE P(X = 3) = \frac{3^3}{3!}e^{-3} = 0.2240$

$\LARGE P(X = 4) = \frac{3^4}{4!}e^{-3} = 0.1680$

$\LARGE P(X = 5) = \frac{3^5}{5!}e^{-3} = 0.1008$

a) The probability here is computed as:

$\LARGE P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

$\LARGE P(X \leq 3) = 0.0498 + 0.1494 + 0.2240 +0.2240$

$\LARGE P(X \leq 3) = 0.6472$

Therefore 0.6472 is the required probability here.

b) The required probability here is computed as:

$\LARGE P(X > 3) = 1 - P(X \leq 3)$

$\LARGE P(X > 3) = 1 - 0.6472 = 0.3528$

Therefore 0.3528 is the required probability here.

c) The required probability here is computed as:

$\LARGE P(2 < X \leq 5) = P(X =3 ) + P(X = 4) + P(X = 5)$

$\LARGE P(2 < X \leq 5) = 0.2240 + 0.1680 + 0.1008$

$\LARGE P(2 < X \leq 5) = 0.4928$

Therefore 0.4928 is the required probability here.

d) We already computed in first part that:

$\LARGE P(X \leq 3) = 0.6472$

$\LARGE P(X \leq 2) = 0.6472 - 0.2240 = 0.4232$

Therefore the smallest value of x required here is 0.4232