Question

PLEASE ANSWER THE HIGHLIGHTED PARTS ONLY. REALLY NEED PART G.

Suppose X has a normal distribution with mean -3 and standard deviation 7. The pnorm(), dnorm() and qnorm() functions should be useful in the following.

a) What is the expected value of 5X + 15?

b) What is the variance of 5X + 15?

c) Calculate the probability X >0.

d) Calculate the probability that X < -10

e) Calculate the probability that 1 < X < 10.

f) If the probability that X > t = .04 then what is t?

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

f) If the probability that X > t = .04 then what is t?

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

Answer 1

(a) $E(5X+15)=5E(X)+15=5 \times -3 + 15=0$

(b) $Var(5X+15)=5^2Var(X)=25 \times 7^2 = 1225$

(c)

Pr(x > 0) = 1 - Pr(x<0) = 1- pnorm(0, -3, 7) = 0.3341176

(d) $Pr(X<-10)=Pr(X\leq -10)=pnorm(-10,-3,7)=0.1586553$

(e)

$Pr(1<X<10)=Pr(X\leq 10)-Pr(X\leq 1) \\ = pnorm(10,-3,7)-pnorm(1,-3,7)= 0.2522092$

(f)

$Pr(X>t)=.04 \\ 1-Pr(X\leq t)=.04\\ Pr(X\leq t)=1-0.04\\ Pr(X\leq t)=0.96\\ t=qnorm(0.96,-3,7)\\ t=9.254802$

(g) qnorm(0.96,-3,7)