Question

2, (3 points) Let X be a standard normal random variable. Let Y = X2. (Use R and give code.) (a) Find P(-1.5 < X < 2.5) (b) Find P(Y1 Notes: . You are not expected to and don't need to figure out the distribution of Y. Just convert the probability for Y to a probability involving X . Algebra review: Use R to find the final results

Please explain your work so I can replicate it and practice. Please also include R code, not just R outputs. Thank you.

Answer 1

X ~ N (0,1)

a) P ( -1.5 < X < 2.5) = P ( X < 2.5) - P ( X < -1.5)

( P ( X <2.5) is cumulative probability at X = 2.5)

By using R

> p1=pnorm(2.5,0,1)-pnorm(-1.5,0,1)
> p1
[1] 0.9269831

P( -1.5 < X < 2.5) = 0.9269831

b) P ( Y > 1) = P ( X² >1)

$= P ( \left | X \right |>1) = P ( X >1 or X < -1)$

= P ( X >1) + P(X <-1)

Since normal distribution is symmetric distribution

P ( X > 1) = P ( X <-1)

Hence P ( Y >1) = 2* P (X <-1)

by using R

> p2 = 2 * pnorm(-1,0,1)
> p2
[1] 0.3173105

P ( Y >1) = 0.3173105

Alternative Solution :

Since Y = X² and X is standard normal variate  , the square of standard normal variate is chi-square variate.

P ( Y > 1) = 1 - P ( Y <1)

by using R

> p3 = 1- pchisq(1,1)
> p3
[1] 0.3173105

P ( Y >1) = 0.3173105.