Chapter 38

Probability

38.1 Improper IntegralsIf f is Riemann integrable on [0,R] for each R, then∫

∞

0f (x)dx≡ lim

R→∞

∫ R

0f (x)dx

if this limit exists. Otherwise the improper integral is not defined. If f is only Riemannintegrable on [δ ,R] for each δ < R, then∫

∞

0f (x)dx≡ lim

(δ ,R)→(0,∞)

∫ R

δ

f (x)dx

provided this limit exists. This expression means: There exists I ≡∫

∞

0 f (x)dx such that foreach ε > 0 there is R0 and δ 0 such that if δ < δ 0 and R > R0, then∣∣∣∣∫ R

δ

f (x)dx− I∣∣∣∣< ε

Otherwise we don’t give a definition of the improper integral. Integrals of the form∫ 0−∞

f (x)dxare defined similarly. As to

∫∞

−∞f (x)dx, it equals∫

∞

0f (x)dx+

∫ 0

−∞

f (x)dx

provided these last two exist. As an application of polar coordinates, here is an importanttheorem.

Theorem 38.1.1∫

∞

0 e−x2dx = 1

2√

π and∫

∞

−∞e−x2

dx =√

π .

Proof: Let IR ≡∫ R

0 e−x2dx. Then IRIR =

∫ R0∫ R

0 e−x2e−y2

dx. Also

I ≡ limR→∞

IR

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