Introduction to Probability and Statistics for Engineers and Scientists

138 Chapter 4:Random Variables and Expectation

46.A machine makes a product that is screened (inspected 100 percent) before being

shipped. The measuring instrument is such that it is difficult to read between 1 and

113 (coded data). After the screening process takes place, the measured dimension

has density

f(z)=







kz^2 for 0≤z≤ 1

1 for 1<z≤ (^113)
0 otherwise
(a) Find the value ofk.
(b)What fraction of the items will fall outside the twilight zone (fall between
0 and 1)?
(c) Find the mean and variance of this random variable.
47.Verify Equation 4.7.4.
48.Prove Equation 4.7.5 by using mathematical induction.
49.LetXhave varianceσx^2 and letYhave varianceσy^2. Starting with
0 ≤Var(X/σx+Y/σy)
show that
− 1 ≤Corr(X,Y)
Now using that
0 ≤Var(X/σx−Y/σy)
conclude that
− 1 ≤Corr(X,Y)≤ 1
Using the result that Var(Z) = 0 implies thatZ is constant, argue that if
Corr(X,Y)=1or−1 thenXandYare related by
Y=a+bx
where the sign ofbis positive when the correlation is 1 and negative when it
is−1.
50.Considernindependent trials, each of which results in any of the outcomesi,i=
1, 2, 3, with respective probabilitiesp 1 ,p 2 ,p 3 ,
∑ 3
i= 1 pi=1. LetNidenote the