Question

In: Math

Suppose W is a standard beta random variable with parameters α=4 and β = 4 which...

Suppose W is a standard beta random variable with parameters α=4 and β = 4 which means W has expected value 4/8 and standard deviation 1/6 Suppose X is a normal random variable with mean 9 and standard deviation 8. Answer the following using R code:
a) Calculate the 61st percentile of the distribution of X

b) Calculate the 98th percentile of the distribution of W.

c) What is the expected value of the random variable -8W - 15?

d) What is the standard deviation of the random variable -8W - 15?

e) What is the standard deviation of the random variable ((x-5)/4)+1?

f) If X and W are independent then what is the variance of 6X - 5W + 5?

g) Copy your R script for the above into the text box here.

Expert Solution

All R script is shown in bold.

a) Calculate the 61st percentile of the distribution of X

qnorm(0.61,9,8)

The answer is 11.23455

b) Calculate the 98th percentile of the distribution of W.

qbeta(0.98,4,4)

The answer is 0.8272883

c) What is the expected value of the random variable -8W - 15?

alpha = 4
beta = 4
mean.w = alpha / (alpha + beta)
-8 * mean.w - 15

The answer is -19

d) What is the standard deviation of the random variable -8W - 15?

Var(-8W - 15) = (-8)² Var(W) + Var(-15) = 64 Var(W)

Standard deviation of -8W - 15 = = 8 sd(w)

sd.w = 1/6
8 * sd.w

The answer is 1.333333

e) What is the standard deviation of the random variable ((x-5)/4)+1?

Var(((x-5)/4)+1) = Var(x -5)/4² + Var(1) = [Var(x) + Var(-5)]/16 = Var(x)/16

standard deviation of the random variable ((x-5)/4)+1 =

sd.x = 8
sd.x / 4

The answer is 2

f) If X and W are independent then what is the variance of 6X - 5W + 5

Var(6X - 5W + 5) = 6² Var(X) + (-5)² Var(W) + Var(5) = 36 Var(X) + 25 Var(W)

sd.w = 1/6
sd.x = 8
36 * sd.x^2 + 25 * sd.w^2

The answer is 2304.694

milcah answered 3 months ago

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