a Find the mean and variance of the fraction of defective mo

(a) Find the mean and variance of the fraction of defective modems per shipment. (You can shift enter f[y] above to define it in Mathematica)

(b) What is the probability that a randomly selected shipment contains at least 30% defective modems?

(c) What is the probability that a randomly selected shipment contains no more than 5% defective modems?

PDF[BetaDistribution[a, b], y] Here, True means Else. Suppose that the fraction of defective modems shipped by a data - communications vendor has a beta distribution with a=4 and b=21. (a) Find the mean and variance of the fraction of defective modems per shipment. (You can shift enter f[y] above to define it in Mathematica) (b) What is the probability that a randomly selected shipment contains at least 30% defective modems? (c) What is the probability that a randomly selected shipment contains no more than 5% defective modems?

Solution

1)for a beta distribution the mean is given by a/(a+b) =4/(21+4)=4/25 =.16

variance is given by=ab/(a+b+1)(a+b)^2    ,here a=4   b=21

variance is =4*21/26*25^2 =.005169    standard deviation =sqrt(variance)=sqrt(.005169)=.07189

b)P(X>=.30)=1-P(X<=.3)

P(X<=.3)

z score=1-(.3-.16)/.07189=1.94

from the z tabgle .9738

required probability is 1-.9738 =.0262

c)P(X<.05) =(.05-.16)/.07189 =-1.53

from the z table value is P(z<-1.53)=.0630