Question

1. Let X and Y be independent random variables with μ_X= 5, σ_X= 4, μ_Y= 2, and σ_Y= 3.

Find the mean and variance of X + Y.

Find the mean and variance of X – Y.

2. Porcelain figurines are sold for $10 if flawless, and for $3 if there are minor cosmetic flaws. Of the figurines made by a certain company, 75% are flawless and 25% have minor cosmetic flaws. In a sample of 100 figurines that are sold, let Y be the revenue earned by selling them and let X be the number of them that are flawless.

Find μYμY.

$

Find σYσY

$

3. Someone claims that a certain suspension contains at least seven particles per mL. You sample 1mL of solution. Let X be the number of particles in the sample.

If the mean number of particles is exactly seven per mL (so that the claim is true, but just barely), what is P(X ≤ 1)? Round the answer to four decimal places.

4. A company receives a large shipment of bolts. The bolts will be used in an application that requires a torque of 100 J. Before the shipment is accepted, a quality engineer will sample 12 bolts and measure the torque needed to break each of them. The shipment will be accepted if the engineer concludes that fewer than 1% of the bolts in the shipment have a breaking torque of less than 100 J.

Assume the 12 values are sampled from a normal population, and assume the sample mean and standard deviation calculated in part (a) are actually the population mean and standard deviation. Compute the proportion of bolts whose breaking torque is less than 100 J. Will the shipment be accepted? Round the answer to two decimal places and express the answer as a percentage.

__% of bolts would have breaking torques less than 100 J.

P(X ≤ 1) =

Answer 1

Answer 1:

μX = 5, σX = 4, μY = 2, and σY = 3

Mean of X + Y = E(X+Y)

E(X+Y) = E(X) + E(Y)

E(X+Y) = μX + μY

E(X+Y) = 5+2

E(X+Y) = 7

Variance of X + Y = Var(X+Y)

Var(X+Y) = Var(X) + Var(Y) + 2*Cov(X,Y)

Var(X) = σX2 = 42 = 16

Var(Y) = σY2 = 32 = 9

Since, X and Y are independent variable so Cov(X,Y) = 0

Substituting all the these values, we get

Var(X+Y) = 16 + 9 + 2*0

Var(X+Y) = 25

Thus, the mean and variance of X + Y is 7 and 25 respectively.

Mean of X - Y = E(X-Y)

E(X-Y) = E(X) - E(Y)

E(X-Y) = μX - μY

E(X-Y) = 5 - 2

E(X-Y) = 3

Variance of X - Y = Var(X-Y)

Var(X-Y) = Var(X) + Var(Y) - 2*Cov(X,Y)

Var(X) = σX2 = 42 = 16

Var(Y) = σY2 = 32 = 9

Since, X and Y are independent variable so Cov(X,Y) = 0

Substituting all the these values, we get

Var(X-Y) = 16 + 9 - 2*0

Var(X-Y) = 25

Thus, the mean and variance of X – Y is 3 and 25 respectively.

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