A chemical supply company ships a certain sol - vent in 10-gallon drums. Let X represent the num - ber of drums ordered by a randomly chosen cus - tomer. Assume X has the following probability mass function:x 0 1 2 3 4p (x) 0.4 0.2 0.2 0.1 0.1 (a) Find the mean number of drums ordered. (b) Find the variance of the number of drums ordered. (c) Find the standard deviation of the number of drums ordered. (d) Let Y be the number of gallons ordered. Find the probability mass function of Y (e) Find the mean number of gallons ordered.

a) 1.3

b) 1.81

c) 1.345

d) Each drum is a 10-gallon drum

In terms of gallons, the probability mass function becomes

y 0 10 20 30 40

p (y) 0.4 0.2 0.2 0.1 0.1

e) 13 gallons

Step-by-step explanation:

a) The mean is given by the expected value for number of drums ordered.

Expected values is given by

E (X) = Σ xᵢpᵢ

x 0 1 2 3 4

p (x) 0.4 0.2 0.2 0.1 0.1

E (X) = (0*0.4) + (1*0.2) + (2*0.2) + (3*0.1) + (4*0.1) = 0 + 0.2 + 0.4 + 0.3 + 0.4 = 1.3

b) Variance = Var (X) = Σx²p - μ²

μ = E (X) = mean

Σx²p = (0² * 0.4) + (1² * 0.2) + (2² * 0.2) + (3² * 0.1) + (4² * 0.1) = 0 + 0.2 + 0.8 + 0.9 + 1.6 = 3.5

Var (X) = Σx²p - μ² = 3.5 - 1.3² = 1.81

c) Standard deviation = √variance = √1.81 = 1.345

d) Each drum is a 10-gallon drum

In terms of gallons, the probability mass function becomes

y 0 10 20 30 40

p (y) 0.4 0.2 0.2 0.1 0.1

e) Mean = E (Y)

E (Y) = Σ yᵢpᵢ

E (Y) = (0*0.4) + (10*0.2) + (20*0.2) + (30*0.1) + (40*0.1) = 13 gallons