1. ## random variable related

1) Let X be a random variable that takes values μ−a, μ, and μ+a with probabilities p, 1 − 2p, and p respectively. Show
Pr(|X − μ| >= a) = Var(X)/a^2

2) For a discrete random variable X, show that m = E(X) minimizes E[(X − m)^2]

2. Originally Posted by ashimb9
1) Let X be a random variable that takes values μ−a, μ, and μ+a with probabilities p, 1 − 2p, and p respectively. Show
Pr(|X − μ| >= a) = Var(X)/a^2
Since \ |x-\mu| \ge a[/tex] when $x=\mu-a$ and $x=\mu+a$ (assuming $a>0$)

$
Pr(|x-\mu| \ge a)=2p
$

You should know how to calculate $Var(X)$ to complete the demonstration.

RonL

3. Originally Posted by CaptainBlack
Since \ |x-\mu| \ge a[/tex] when $x=\mu-a$ and $x=\mu+a$ (assuming $a>0$)

$
Pr(|x-\mu| \ge a)=2p
$

You should know how to calculate $Var(X)$ to complete the demonstration.

RonL
But how can it be assumed for certain that a>0 ?

4. Originally Posted by ashimb9
But how can it be assumed for certain that a>0 ?
If $a<0$, then it is always true that $|x-\mu|>a$ since $|x-\mu|$ is always positive.

5. Originally Posted by ashimb9
But how can it be assumed for certain that a>0 ?
You will probabbly find it as an extra condition somewhere, because the question makes little sense otherwise.

RonL