Let Z denote astandardnormalrandomvariable,whichhas pdf (z) = (1~ 2) exp(z2~2) and cdf . Recallthat Y = Z2 has a 21 distribution. (a) Explainwhythe cdf

Let Z denote astandardnormalrandomvariable,whichhas pdf ϕ(z) = (1~

º

2π) exp(−z2~2)

and cdf Φ. Recallthat Y = Z2 has a χ21 distribution.

(a) Explainwhythe cdf of Y for y ≥ 0 is F(y) = Φ(º

y) − Φ(−º

y).

(b) Takingthederivative, showthatthe pdf of Y is24 f(y) = (1~

º

y)[ϕ(

º

y) + ϕ(−

º

y)] = (1~

»

2πy)e−y~2, y ≥ 0.

Using Γ(1~2) =

º

π, thisisthegamma pdf (2.10) with λ = 1~2 and k = 1~2.