Statistics Fall 2004 Theory of Probability Practice Final # 1 { Solutions
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1 Statistics 6 - Fall 4 Theor of Probabilit Practice Fial # { Solutios Istructios Aswer Q. -6. All questios have equal weight. Q. Let X ad Y be idepedet, both with Geometric distributio Geom(p For itegers i; with i <, compute Solutio For i P (X ijx + Y, we have P (X i; X + Y P (X ijx + Y P (X + Y P (X i; Y i P (X + Y p( pi p( p i p ( p Q. Let X ad Y be discrete iteger valued radom variables with joit p.m.f. give b p(; p ( p e ; ; (a Compute E(X. (b Compute E(Y. (c Compute Cov(X; Y. Solutio (a For (a, a little calculatio shows that the margial p.m.f. of X is quite mess. I this case, we ca either use the epressio E(X E(E(XjY
2 otig that XjY Biomial(; p ad Y Poisso( so that Or, more directl (b Similarl (c Fiall ad E(X E(X E(E(XjY E(pY p E(Y p X X X X p X X e X X e p p ( p e X p ( p e X e e p ( p e X p ( p Cov(X; Y E(XY E(XE(Y E(XY p E(XY p X X X X X p ( p e p ( p e X e p p( + e
3 Therefore, Cov(X; Y p Alterativel, istead of doig the double sum, we ca use the fact that E(XY jy Y E(XjY ; so that E(XY E(E(XY jy E(Y E(XjY E(pY p( + Q. 3 Let (X be a sequece of Biomial radom variables such that X Biom(; p Set Y X p p (a Show directl that M Y (t (b Show b direct calculatio that pe ( ptp + ( pe ptp lim M Y (t e t p( p Solutio (a We kow, b the Biomial theorem, that X M X (t e t p ( ( p + pe t p Therefore, M X p(t e pt ( p + pe t ( pe tp + pe t( p ad M Y (t M (X p p (t M X p(t p (b Takig logs of both sides, it is eough to show that lim log M Y (t t p( p ( pe tpp + pe t( pp 3
4 Sedig lim log M Y (t lim log ( pe tpp + pe t( pp ( pe tpp + pe t( pp log lim lim lim ( ppt ( ppt ( ppt 3 (e tpp e t( pp (( pe tpp +pe t( pp ( ppt(e tpp e t( pp (( pe tpp +pe t( pp lim p (pt 3 e tpp +( pt 3 e t( pp (( pe tpp +pe t( pp 3 (e tpp e t( pp p( pt 3 (( pe tpp +pe t( pp lim 3 (pt 3 e tpp +( pt 3 e t( pp (( pe tpp +pe t( pp 3 C A ( ppt (pt + ( ( ppt pt A simpler wa is to use the Talor series epasio for the fuctio f( e which ields e tpp + p tp + t p + e t e t( pp + pp t3 p t( p p + t ( p + e t ( pp t3 ( p for t ; t t (assumig t >, otherwise reverse iequalities. Therefore, ( pe tpp + pe t( pp ' with the error of the order 3 + t p( p 4
5 This meas M Y (t ' + t p( p e t p( p Q. 4 Let (X ; ; X be idepedet epoetial radom variables havig a commo parameter. (a Compute P ma i X i log + (b Show that, for ever, lim P ma X i log + i e e Solutio (a Note that ma X i log + i B idepedece P \ i fx i log + g \ i Y i fx i log + g P (X i log + P (X i log + e log e (b lim e e e Q. 5 Oe has lightbulbs whose lifetimes are idepedet epoetial radom variables with mea 5 hours. If the bulbs are used oe at a time, with a failed bulb replaced immediatel b a ew oe, what is the probabilit that there is still a workig bulb after 55 hours? Solutio The evet that there is still a lightbulb workig after 55 hours is the evet that there were 55 total hours of use i the lightbulbs iitiall. More formall, let X i be the lifetime of the i-th lightbulb, the fworkig bulb after 55 hoursg 5 ( X i X i > 55
6 Applig the Cetral Limit Theorem, P i X i is approimatel ormall distributed with mea 5 ad variace 5 (remember, if the mea of the epoetial is 5, the 5 ad the variace is therfore 5 { addig up idepedet epoetials with parameter /5 there gives variace 5. Or, P i X i 5 p 5 P i X i 5 5 is approimatel a stadard ormal radom variable. Therefore, X P i P X i > 55 P X i > 5 5 i P i P X i 5 > 5 5 ' 38 (5 Q. 6 If X ad Y are idepedet cotiuous positive radom variables, epress the desit fuctio of Z XY as a fuctio of the desities of X ad Y. Evaluate this epressio i the special case where X ad Y are both epoetial radom variables with parameter. Solutio The distributio fuctio of Z is give b F Z (z P (Z z P (XY z P ((X; Y f(; zg ZZ f(;zg Z Z z Z Z Z Z z f X ( f X (( Therefore, the desit of Z is give b f Z (z d dz Z f X (f Y ( d d f X (f Y ( d d f X (f Y ( d d Z z f X (( F Y (z d f Y ( d F Y (z d Z d z f X(f Y (z d 6
7 I the special case that X ad Y are epoetial with parameter, this desit is f Z (z Z Z z e (+z d z ue u(+z du Z z + z z ( + z (z + ( + zue u(+z du 7
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