Problem set 9, submit solutions by 14.11.2012

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Prof. N. Kajino, Probability Theory WS 2012/2013

Problem set 9, submit solutions by 14.11.2012

TheProblems below will be discussed in the tutorial on 16.11.2012.

(TheExerciseis additional and will be discussed only if time permits.) In the problems and the exercise below,.;F;P/denotes a probability space and all random variables are assumed to be defined on.;F;P/.

Problem 3.1. Letd 2Nand letx2 R^d. Prove that the unit massıx atxdefined by ıx.A/WD1A.x/,A2B.R^d/(recall Example 1.5-(2)), does not have a density.

Problem 3.2. CalculateEŒX and var.X /for a real random variableXwith (1) the binomial distributionB.n; p/,n2N,p2Œ0; 1.

(2) the Poisson distribution Po./,2.0;1/.

(3) the geometric distribution Geom.˛/,˛2Œ0; 1/.

Problem 3.3. CalculateEŒX and var.X /for a real random variableXwith (1) the uniform distribution Unif.a; b/,a; b2R,a < b.

(2) the exponential distribution Exp.˛/,˛2.0;1/.

(3) the gamma distribution Gamma.˛; ˇ/,˛; ˇ2.0;1/.

Problem 3.4. LetX be an exponential random variable. Prove that

PŒX > sCt jX > sDPŒX > t for anys; t 2Œ0;1/ (3.79) (recall (1.66) for the definition of conditional probabilities).

(3.79) is known as the “memoryless property” of exponential random variables.

Due to this property, exponential random variables are often used as“random alarm clocks with no memory”.

Exercise 3.5. LetX be a real random variable such thatPŒX > 0 > 0, and suppose PŒX > sCt jX > s DPŒX > t for anys; t 2 .0;1/withPŒX > s > 0. Define hWR!Œ0; 1byh.t /WDPŒX > t . Prove the following statements:

(1)his right-continuous andh.sCt /Dh.s/h.t /for anys; t 2Œ0;1/.

(2) There exists˛2.0;1/such thath.t /De ^˛tfor anyt 2Œ0;1/.

(3)Xis an exponential random variable of parameter˛.

Problem 3.6. LetX be a normal random variable with meanm and variance v 2 .0;1/. Prove that the real random variableY WDe^X has a densityY given by

Y.x/D 1 xp

2vexp

.logx m/²

2v

1.0;1/.y/: (3.80) The law ofY is called thelognormal distribution with parametersm; v.

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Problem 3.7. LetX be a normal random variable with mean0and variance1. Prove that the real random variableZWDX²has a densityZgiven by

Z.x/D 1

p2xe ^x=21.0;1/.x/: (3.81) The law of Z is called the chi square distribution with one degree of freedomand denoted as²₁. (In fact, (3.81) and (3.21) easily imply that²₁DGamma.1=2; 1=2/.) Problem 3.8. Letm 2 R,˛ 2 .0;1/and letX be a Cauchy random variable with parametersm; ˛. Prove thatXdoes not admit the mean, i.e.EŒX^CDEŒX D 1.

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