Question

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.

Answer 1

Answer:

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