How can i calculate the given events? (details inside, a bit complicated for me)

let y=(x-z)/2, where x is a normal random variable with mean z and variance of 4.

how can i calculate:
1)P{-2 \leq x-z \leq 4}

2)what is E[Y^2]

3)how can i calculate p{Y \leq a | B} (for every a \in RR if known that B={|x-z| \leq 2} occured?

if you can, please explain how you solved. i find this question very difficult and would appreciate learning how to solve it

1 Answer
May 5, 2018

"See explanation"

Explanation:

"y is standard normal (with mean 0 and standard deviation 1)"
"So we use this fact."

"1) "= P[ - 1 <= (x-z)/2 <= 2 ]
"We now look up the z values in a table for z values for"
"z = 2 and z = -1. We get"
0.9772" and "0.1587.
#=> P = 0.9772 - 0.1587 = 0.8185

"2) "var = E[x^2] - (E[x])^2
=> E[x^2] = var + (E[x])^2
"Here we have var = 1 and mean = E[Y] = 0."
=> E[Y^2] = 1 + 0^2 = 1

"3) "P[ Y <= a | B ] = (P[ Y <= a" AND "B ]) / (P[ B ])
P[ B ] = 0.8413 - 0.1587 = 0.6826 " (z values table)"
P[ Y <= a " AND "B] = 0, " if "a < -1
P[ Y <= a " AND "B] = P[ -1 <= Y <= a ]" , if "-1 <=a <= 1
P[ Y <= a " AND "B] = P[B]" , if "a > 1

=> P[ Y <= a | B ] = 0, " if "a < - 1
=> P[ Y <= a | B ] = (T(a) - 0.1587)/0.6826, " if "-1 <= a <= 1
"(with T(a) the value in z values table for z=a)"
=> P[ Y <= a | B ] = 1, " if "a > 1