Probability Distributions Binomial show all formulas and wo

Probability Distributions & Binomial (show all formulas and work)

1. Find the mean and Standard deviation for the probability distribution.

2. At a college, 40% of all students are full-time.

a) If we randomly select 10 students what is the probability that exactly 5 are full-time?

b) If we randomly select 75 students, what is the expected value and standard deviation for the number of full-time students?

Solution

1.

Mean = X*p(X) = 5*0.10+6*0.18+7*0.21+8*0.25+9*0.26 = 7.39

Variance = X²p(X) - Mean² = 5²*0.10+6²*0.18+7²*0.21+8²*0.25+9²*0.26 - 7.39² = 1.7179

Standard Deviation = sqrt(Var) = sqrt(1.7179) = 1.31

2.

Let p denote the probability that a student is full time.

p = 0.4

(a)

Out of 10 students which have to be selected exactly 5 have to be full time.

P(5 out of 10 students are full time ) = C(10,5)*0.4⁵(1-0.4)^10-5 = 0.20

(b)

Expected value of binomial distribution is n*p = 75*0.4 = 30

Standard deviation of binomial distribution is sqrt(n*p*(1-p)) = sqrt(75*0.4*0.6) = 4.24