Examples

Expectation of function on RV given by chart

Suppose that in such a way that and and and no other values are mapped to .

	1	2	3
	4	1	87

Then:

And:

Therefore:

Variance of uniform random variable

The uniform random variable on has distribution given by when .

(a) Find using the shorter formula.

(b) Find using “squaring the scale factor.”

(c) Find directly.

Solution (a)

(1) Compute density.

The density for is:

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(2) Compute and directly using integral formulas.

Compute :

Now compute :

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(3) Find variance using short formula.

Plug in:

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(b)

(1) “Squaring the scale factor” formula:

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(2) Plugging in:

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(c)

(1) Density.

The variable will have the density spread over the interval .

Density is then:

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(2) Plug into prior variance formula.

Use and .

Get variance:

Simplify:

PDF of derived from CDF

Suppose that .

(a) Find the PDF of . (b) Find the PDF of .

Solution

(a)

Formula:

Plug in:

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(b)

By definition:

Since is increasing, we know:

Therefore:

Then using differentiation:

Probabilities via CDF

Suppose the CDF of is given by . Compute:

(a) (b) (c) (d)

Solution