3.4. The Normal Distribution 1953.4.8.Evaluate∫ 3
2 exp[−2(x−3)(^2) ]dx.
3.4.9.Determine the 90th percentile of the distribution, which isN(65,25).
3.4.10.Ife^3 t+8t
2
is the mgf of the random variableX, findP(− 1 <X<9).
3.4.11.Let the random variableXhave the pdf
f(x)=
2
√
2 π
e−x
(^2) / 2
, 0 <x<∞, zero elsewhere.
(a)Find the mean and the variance ofX.
(b)Find the cdf and hazard function ofX.
Hint for (a): ComputeE(X) directly andE(X^2 ) by comparing the integral with
the integral representing the variance of a random variable that isN(0,1).
3.4.12.LetXbeN(5,10). FindP[0. 04 <(X−5)^2 < 38 .4].
3.4.13.IfXisN(1,4), compute the probabilityP(1<X^2 <9).
3.4.14.IfXisN(75,25), find the conditional probability thatXis greater than
80 given thatXis greater than 77. See Exercise 2.3.12.
3.4.15.LetX be a random variable such thatE(X^2 m)=(2m)!/(2mm!),m=
1 , 2 , 3 ,...andE(X^2 m−^1 )=0,m=1, 2 , 3 ,.... Find the mgf and the pdf ofX.
3.4.16.Let the mutually independent random variablesX 1 ,X 2 ,andX 3 beN(0,1),
N(2,4), andN(− 1 ,1), respectively. Compute the probability that exactly two of
these three variables are less than zero.
3.4.17. Compute the measures of skewness and kurtosis of a distribution which
isN(μ, σ^2 ). See Exercises 1.9.14 and 1.9.15 for the definitions of skewness and
kurtosis, respectively.
3.4.18.Let the random variableXhave a distribution that isN(μ, σ^2 ).
(a)Does the random variableY=X^2 also have a normal distribution?
(b)Would the random variableY =aX+b,aandbnonzero constants have a
normal distribution?
Hint: In each case, first determineP(Y≤y).
3.4.19.Let the random variableXbeN(μ, σ^2 ). What would this distribution be
ifσ^2 =0?
Hint: Look at the mgf ofXforσ^2 >0 and investigate its limit asσ^2 →0.
3.4.20.LetY have atruncateddistribution with pdfg(y)=φ(y)/[Φ(b)−Φ(a)],
fora<y<b, zero elsewhere, whereφ(x)andΦ(x) are, respectively, the pdf and
distribution function of a standard normal distribution. Show then thatE(Y)is
equal to [φ(a)−φ(b)]/[Φ(b)−Φ(a)].