Fall 2007 Math 221 Final Review Solutions
Emily King T.A.
December 11, 2007
12.4 : Normal Random Variables
1. Let Xbe a random variable with probability density function
f(x) = 1
σ√2πe−
1
2[x−µ
σ]2
Then we call Xis a normal random variable and fis a normal density
function. Also E(X) = µand Var(X) = σ2. If µ= 0 and σ= 1 then we
say it is standard normal and denote the random variable with a Z.
2. Let Zbe a standard normal random variable. Then
•If z > 0, Pr(Z≤z) = Pr(−∞ < Z ≤0) + Pr(0 ≤Z≤z) =
.5 + Pr(0 ≤Z≤z).
•If z < 0, Pr(z≤Z≤0) = Pr(0 ≤Z≤ −z).
•If z > 0, Pr(Z≥z) = Pr(0 ≤Z < ∞)−Pr(0 ≤Z≤z) = .5−Pr(0 ≤
Z≤z).
3. If Xis a normal random variable with mean µand standard deviation σ,
then for any a,b,
Pr(a≤X≤b) = Pr(a−µ
σ≤Z≤b−µ
σ)
where Zis the standard normal random variable. You use all of these
facts to compute probabilities.
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