Question

Round all your answers to 4 decimal places. Let μ=E(X), σ=stanard deviation of X. Find the probability P(μ-σ ≤ X ≤ μ+σ) if X has...

a. ... a Binomial distribution with n=6 and p=1/3.

b. ... a Geometric distribution with p = 0.63.

c. ... a Poisson distribution with λ = 7.2.

Answer 1

Solution :

Given that,

P( - 1< X < + 1)

(a)

n = 6

p = 1/3 = 0.3333

q = 1 - p = 1 - 0.3333 = 0.6667

Using binomial distribution,

= mean = n * p = 6 * 0.3333 = 1.9998

standard deviation = n * p * q = 6 * 0.3333 * 0.6667 = 1.1547

= 1.1547

P( - 1< X < + 1) =

P(1.9998 - 1.1547 < X < 1.9998 + 1.1547)

= P(0.8451 < X < 3.1545)

(b)

p = 0.63

= (1 - 0.63) / 0.63 = 0.5873

= (1 - 0.63) / 0.63² = 0.9655

P(0.5873 - 0.9655 < X < 0.5873 + 0.9655)

= P(0 < X < 1.5528)

(c)

= 7.2

= 2.6833

P(7.2 - 2.6833 < X < 7.2 + 2.6833)

= P(4.5167 < X < 9.8833)