Write a program in R: Let X be a normally distributed random variable with mean

Question

Write a program in R:

Let X be a normally distributed random variable with mean 2 and variance 9.

(a) Let Y = 5X + 2, what is the distribution of Y? What are it’s mean and variance?

(b) Find P(Y < 10) and P(X < 10).

(c) What value of Y marks the 67th percentile?

(d) The 30th percentile of the standard normal distribution is 0.5244. How can you use this information to find the value of X that marks the 30th percentile?

(e) Use a normal table to calculate the answers for parts (b)-(c). Make sure your answers match.

Explanation / Answer

Ans:

If X is normally distributed with parameters µ and 2 , then the random variable Y = aX + b is normally distributed with parameters aµ + b and a22 .

a)Y=5X+2

Distribution of Y:

mean=5*2+2=12

variance=25*9=225

standard dev.=sqrt(225)=15

b)P(Y<10)

z=(10-12)/15=-2/15=-0.13

P(Y<10)=P(z<-0.13)=0.4483

P(X<10)

z=(10-2)/3=8/3=2.67

P(X<10)=P(z<2.67)=0.9962

c)

P(Z<=z)=0.67

NORMSINV(0.67)=0.44

y=12+0.44*15=18.6

d)

P(Z<=z)=0.3

NORMSINV(0.3)=-0.524

x=2-0.524*3=0.428

Use pnorm for cumulative probability and qnorm for inverse of cdf as given below:

qnorm(0.67, 0, 1)=0.44

qnorm(0.3, 0, 1)=-0.524

pnorm(-0.13, 0, 1)=0.4483

pnorm(2.67, 0, 1)=0.9962

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