Question

Given a random variable X following normal distribution with mean of -3 and standard deviation of 4. Then random variable Y=0.4X+5 is also normal.

(1)Find the distribution of Y, i.e. μy,σy

(2)Find the probabilities P(−4<X<0),P(−1<Y<0)

(3)Find the probabilities P(−4<X¯<0),P(3<Y¯<4)

(4)Find the 53th percentile of the distribution of X

Answer 1

#Que 1

We know that if X~ N( μ ,σ^2) i.e with mean μ and standard deviation σ

then Y=aX+b~N(aμ+b,(aσ)^2) i.e with mean aμ+b and standard deviation aσ

here X~N(-3,16)

hence Y=0.4X+5~N(-1.2,2.56) i.e with mean -1.2 and standard deviation 2.56

#Que 2

P(−4<X<0)= P(X<0)-P(X<-4)

> pnorm(0,-3,4)-pnorm(-4,-3,4)
[1] 0.372079
> P(−1<Y<0)=> pnorm(0,-1.2,1.6)-pnorm(-1,-1.2,1.6)
[1] 0.2236344

P(3<Y<4)=> pnorm(4,-1.2,1.6)-pnorm(3,-1.2,1.6)
[1] 0.003755423

#Que 4

> qnorm(0.53,-3,4)
[1] -2.698921