2.8. Linear Combinations of Random Variables 153Example 2.8.2(Sample Variance).Define thesample variancebyS^2 =(n−1)−^1∑ni=1(Xi−X)^2 =(n−1)−^1(n
∑i=1Xi^2 −nX
2)
, (2.8.7)where the second equality follows after some algebra; see Exercise 2.8.1.
In the average that defines the sample varianceS^2 , the division is byn− 1
instead ofn. One reason for this is that it makesS^2 unbiased forσ^2 ,asnext
shown. Using the above theorems, the results of the last example, and the facts
thatE(X^2 )=σ^2 +μ^2 andE(X
2
)=(σ^2 /n)+μ^2 , we have the following:
E(S^2 )=(n−1)−^1(n
∑i=1E(Xi^2 )−nE(X2
))=(n−1)−^1{
nσ^2 +nμ^2 −n[(σ^2 /n)+μ^2 ]}= σ^2. (2.8.8)Hence,S^2 is unbiased forσ^2.EXERCISES2.8.1.Derive the second equality in expression (2.8.7).2.8.2.LetX 1 ,X 2 ,X 3 ,X 4 be four iid random variables having the same pdff(x)=
2 x, 0 <x<1, zero elsewhere. Find the mean and variance of the sumY of these
four random variables.
2.8.3.LetX 1 andX 2 be two independent random variables so that the variances
ofX 1 andX 2 areσ 12 =kandσ^22 = 2, respectively. Given that the variance of
Y=3X 2 −X 1 is 25, findk.
2.8.4. If the independent variablesX 1 andX 2 have meansμ 1 ,μ 2 and variances
σ^21 ,σ^22 , respectively, show that the mean and variance of the productY =X 1 X 2
areμ 1 μ 2 andσ 12 σ^22 +μ^21 σ^22 +μ^22 σ 12 , respectively.
2.8.5.Find the mean and variance of the sumY=∑ 5
i=1Xi,whereX^1 ,...,X^5 are
iid, having pdff(x)=6x(1−x), 0 <x<1, zero elsewhere.
2.8.6. Determine the mean and variance of the sample meanX =5−^1
∑ 5
i=1Xi,
whereX 1 ,...,X 5 is a random sample from a distribution having pdff(x)=4x^3 , 0 <
x<1, zero elsewhere.
2.8.7.LetXandY be random variables withμ 1 =1,μ 2 =4,σ^21 =4,σ 22 =
6 ,ρ=^12. Find the mean and variance of the random variableZ=3X− 2 Y.2.8.8. LetX andY be independent random variables with meansμ 1 ,μ 2 and
variancesσ^21 ,σ^22. Determine the correlation coefficient ofXandZ=X−Y in
terms ofμ 1 ,μ 2 ,σ^21 ,σ^22.