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(let n size =8) P(−4<X¯<0),P(3<Y¯<4)

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

Answer 1

Answer:

Given Data

A random variable X following normal distribution

Mean =  -3

Standard deviation = 4

Random variable Y=0.4X+5 is also normal distribution

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

  

= -1.2 + 5

= 3.8

= 2.56

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

P(−1<Y<0)

=  

= I(-2.375)-I(2.375)

= I(3) - I(2.375)

= 0.998 - 0.991

= 0.007

3)Find the probabilities(let n size =8)

=

=

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

  

2.68