111 AT&T; 8:42 PM utsa.blackboard.com 2. (15 points) Professor A is planning the

Question

111 AT&T; 8:42 PM utsa.blackboard.com 2. (15 points) Professor A is planning the big Statistics Department Super Bowl party Statisticians take pride in their variability, and it is not certain what kinds of chips people will choose. Professor A decides to flip a fair coin to help her make the decision. There are 4 different kinds of chips in the aisle, for each kind of chip, she flips the fair coin independently. If the result is head up, she chooses that kind of chip, otherwise she will not choose. Let the random variablex number of kinds of chips she chosen. (I) Construct the probability distribution table of the random variable x, is x a specific discrete random variable we introduced in class? (5 points) (2) What is the mean of the random variable x? (5 points) (3) What is the standard deviation of the random variable x? (5 points)

Explanation / Answer

(1)

Here, X is random variable which represents number of kinnds of chips professor chosen out of n=4 chips & probability of selecting each chip is p=0.5 (because coin is fair ,probability of head is 0.5) & all coins are fliped independently.

Clearly , X ~Binomial(n=4,p=0.5)

Probability mass function of X is ,

P(X=x)=nCx*px*(1-p)n-x=4Cx*0.5x*0.54-x ;x=0,1,2,3,4

(2) Mean of r.v. X is ,

E(X)=n*p=4*0.5=2

(3)Standard deviation of r.v. X is ,

SD(X)=sqrt(n*p*(1-p))=sqrt(4*0.5*0.5)=1

X P(X=x) 0 4C0*0.50*0.54-0 =0.0625 1 4C1*0.51*0.54-1 =0.25 2 4C2*0.52*0.54-2 =0.375 3 4C3*0.53*0.54-3 = 0.25 4 4C4*0.54*0.54-4 =0.0625 Total

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