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4.1.4 Solved Problems:
Continuous Random Variables
Problem
Let X be a random variable with PDF given by
fX(x)={cx20|x|≤1otherwise
Find the constant c.
Find EX and Var(X).
Find P(X≥12).
Solution
Problem
Let X be a continuous random variable with PDF given by
fX(x)=12e−|x|,for all x∈R.
If Y=X2, find the CDF of Y.
Solution
First, we note that RY=[0,∞). For y∈[0,∞), we have
FY(y) =P(Y≤y)
=P(X2≤y)
=P(−y√≤X≤y√)
=∫y√−y√12e−|x|dx
=∫y√0e−xdx
=1−e−y√.
Thus,
FY(y)={1−e−y√0y≥0otherwise
Problem
Let X be a continuous random variable with PDF
fX(x)={4x300<x≤1otherwise
Find P(X≤23|X>13).
Solution
Problem
Let X be a continuous random variable with PDF
fX(x)={x2(2x+32)00<x≤1otherwise
If Y=2X+3, find Var(Y).
Solution
Problem
Let X be a positive continuous random variable. Prove that EX=∫∞0P(X≥x)dx.
Solution
Problem
Let X∼Uniform(−π2,π) and Y=sin(X). Find fY(y).
Solution
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Introduction to Probability by Hossein Pishro-Nik is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License
