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find the expectation e(x) and variance var(x) for the task
suppose you choose a real number x from the interval [2, 10] with a density function of the form f(x) = c/x , where c is a constant

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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

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