September2820111.Theprobabiltyofbeingdealtafullhouseinpokerisapproximately.0014.

September2820111.Theprobabiltyofbeingdealtafullhouseinpokerisapproximately.0014.Findapproximatelytheprobabilitythatin1000handsofpokeryouwillbedealtatleast2fullhouses.2.Sup
HW due Tuesday 10/4 September 28 2011 1. The probabilty of being dealt a full house in poker is approximately .0014. Find approximately the probability that in 1000 hands of poker you will be dealt at least 2 full houses. 2. Suppose that X has a continuous density f > 0. Show that the median is the value of a which miniminzesE(jX ?? aj) Remark: the assumptions on the density are just to make sure the median is uniquely determined. 3. Assume all second moments below are nite. (a) Let X and Xbe i.i.d.1 r.v.s. Show that 2(X) = 12E((X ?? X)2) (b) Let (XY) be a pair of jointly distributed random variables. Suppose if you like that they have a joint density f(x; y). Let (X; Y ) be a pair of random variables with the same joint distribution but independent of (X; Y ). Show that cov(X; Y ) = 12E((X ?? X)(Y ?? Y )) (c) Use that result to show that if h is a monotonically increasing func- tion the X and h(X) are positively correllated. Hint: apply the foregoing to the pair (X; h(X)). 4. Let A and B be two events. Show that IA; IB are positively correlated i2 P(AjB) > P(A) 5. (a) Let X1;X2;X3 be i.i.d. U(01) random variables. Find the distribu- tion of X1X2X3. (b) XQ1;X2; :::;X9 be i.i.d. U(01) random variables. Find the distribution of 9i=1 Xi: 1independent identically distributed 2i= if and only if 1
Attachments: