Question

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

Answer 1

`"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`