Question

For each of the following random variable, write down the probability dis-tribution (with the distribution family and parameters), then calculate theexpected value and variance of the random variable.

(a) (4 points) John and Fred agree to play a series of tennis practice games.The practice ends whenFredwins three games. Suppose that theprobability forJohnto win each game is 0.6. Random variableXdenote the number of games thatJohnwins before the practice ends.

(b) (4 points) A student has a drawer of ten AA batteries. However, thestudent does not remember that three of the ten batteries have losttheir charge (will not work). She randomly grabs four batteries andbring them with her on travel. Random variableYdenote the numberof batteries she bring on travel that have lost their charge.

(c) (4 points) People withO−blood are called universal donors bacausethey may give blood to anyone without risking incompatibility dueto blood type factors (A and B) or to the Rh factor. Of the peopledonating blood at a clinic, 9% haveO−blood. Random variableZdenote the number of blood donors who are notO−before the firstO−donor arrives on a random day.

Answer 1

Solution: ca) bet x dendes -the umbemes -that John ustnrs et x denotes the numbeo actice en ee ame 91-I t 3-1 nejative binomial dStxtbutior om the ofely of Ne Ne 1 E(x) - Pi (I-P) I-016 '. E(X) - 4i5

1-06) (b) Grven -hat -hiee oj -the ten batlext es have losk set denles numbe batteites that have lost -theis chae aong -the H battesies V) 0 3 (1-0.3 4- O. loey o

V(y)- np (i-P) (C> G ven -that clinic have 0- blood 2 lenotes the umber f blood dondls aho ase mot o don8 befie -the flst don aves ona andom da eome t9.c E(2)

I009 0.09 E 2)to,1 and v(2) 1-p pa = 10:09) 2 v12) = 112.34568