Let Z denote astandardnormalrandomvariable,whichhas pdf (z) = (1~ 2) exp(z2~2) and cdf . Recallthat Y =
Question:
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.
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Related Book For
Foundations Of Statistics For Data Scientists With R And Python
ISBN: 9780367748456
1st Edition
Authors: Alan Agresti
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