Question

Please show all work, step by step, so I can understand what to do.

Ten trials are conducted in a Bernoulli process in which the probability of success in a given trial is 0.3.

If x = the number of successes, determine the following:

a. E(x) b. σx c. P(x = 5) d. P(4 ≤ x ≤ 8) e. P(x > 4)

Answer 1

This is Bernoulli process with 10 trial with probability 0.3 for each trial.

Therefore,

X ~ Bin(n,p)

Where n = 10 , p = 0.3

P(X) = ⁿC_x * p^X * ( 1 - p)^n-x

a)

E(X) = np

= 10 * 0.3

= 3

b)

x = sqrt [ np (1-p) ]

= sqrt [ 10 * 0.3 * 0.7 ]

= 1.4491

c)

P( X = 5) = ¹⁰C₅ * 0.30⁵ * ( 1 - 0.30)⁵

= 0.1029

d)

P(4 <= X <= 8) = P(X = 4) +P(X = 5) +P(X = 6) +P(X = 7) +P(X = 8)

= ¹⁰C₄ * 0.30⁴ * ( 1 - 0.30)⁶ +¹⁰C₅ * 0.30⁵ * ( 1 - 0.30)⁵ +¹⁰C₆ * 0.30⁶ * ( 1 - 0.30)⁴ +¹⁰C₇ * 0.30⁷ * ( 1 - 0.30)³

+¹⁰C₈ * 0.30⁸ * ( 1 - 0.30)²

= 0.3503

e)

P(X > 4) = 1 - P(X <= 4)

= 1 - [ P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) ]

=1 - [ ¹⁰C₀ * 0.30⁰ * ( 1 - 0.30)¹⁰ +¹⁰C₁ * 0.30¹ * ( 1 - 0.30)⁹ +¹⁰C₂ * 0.30² * ( 1 - 0.30)⁸ + ¹⁰C₃ * 0.30³ * ( 1 - 0.30)⁷

+ ¹⁰C₄ * 0.30⁴ * ( 1 - 0.30)⁶ ]

= 0.1503