Let Y = |Z|, where Z ∼ N (0, 1) be a discrete random variable with the following PMF: (i) Find E(Y ) and V ar(Y ). (ii) Find V ar(Y ). (iii) Find the CDF and PDF of Y .
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Expert's answer
2022-01-11T17:28:21-0500
The folded normal distribution is a probability distribution related to the normal distribution. Given a normally distributed random variable X with mean μ and variance σ2, the random variable Y=∣X∣ has a folded normal distribution.
The mean of the folded distribution is
μY=σπ2e−μ2/(2σ2)+μ[1−2Φ(−σμ)]
Given Z∼N(0,1),Y=∣Z∣. Then μ=0,σ=1
μY=1π2e−0+0[1−2Φ(−0)]=π2
(i)
E(Y)=μY=π2
(ii)
The variance is expressed in terms of the mean:
σY2=μ2+σ2−μY2
Given Z∼N(0,1),Y=∣Z∣. Then μ=0,σ=1
σY2=02+12−(π2)2=1−π2=ππ−2
(iii)
The probability density function (PDF) is given by
fY(y;μ,σ2)=2πσ21e−(y2−μ2)/(2σ2)
+2πσ21e−(y2+μ2)/(2σ2)
for y≥0, and everywhere else.
Given Z∼N(0,1),Y=∣Z∣. Then
fY(y;0,1)=2π(1)21e−(y2−02)/(2(1)2)
+2π(1)21e−(y2+02)/(2(1)2)
fY(y;0,1)=π2e−y2/2
for y≥0, and everywhere else.
In our case Y=∣Z∣ follows a half-normal distribution.
The cumulative distribution function (CDF) is given by
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