1.7. Continuous Random Variables 59(b)F(x)=exp{−e−x},−∞<x<∞.(c)F(x)=(1+e−x)−^1 ,−∞<x<∞.(d)F(x)=∑x
j=1( 1
2)j
.1.7.12.Find the cdfF(x) associated with each of the following probability density
functions. Sketch the graphs off(x)andF(x).(a)f(x)=3(1−x)^2 , 0 <x<1, zero elsewhere.(b)f(x)=1/x^2 , 1 <x<∞, zero elsewhere.(c)f(x)=^13 , 0 <x<1or2<x<4, zero elsewhere.Also, find the median and the 25th percentile of each of these distributions.1.7.13.Consider the cdfF(x)=1−e−x−xe−x, 0 ≤x<∞, zero elsewhere. Find
the pdf, the mode, and the median (by numerical methods) of this distribution.
1.7.14.LetXhave the pdff(x)=2x, 0 <x<1, zero elsewhere. Compute the
probability thatXis at least^34 given thatXis at least^12.
1.7.15.The random variableX is said to bestochastically largerthan the
random variableY if
P(X>z)≥P(Y>z), (1.7.14)
for all realz, with strict inequality holding for at least onezvalue. Show that this
requires that the cdfs enjoy the following property:
FX(z)≤FY(z),for all realz, with strict inequality holding for at least onezvalue.
1.7.16.LetXbe a continuous random variable with support (−∞,∞). Consider
the random variableY =X+Δ, where Δ>0. Using the definition in Exercise
1.7.15, show thatY is stochastically larger thanX.
1.7.17.Divide a line segment into two parts by selecting a point at random. Find
the probability that the length of the larger segment is at least three times the
length of the shorter segment. Assume a uniform distribution.
1.7.18.LetXbe the number of gallons of ice cream that is requested at a certain
store on a hot summer day. Assume thatf(x)=12x(1000−x)^2 / 1012 , 0 <x<1000,
zero elsewhere, is the pdf ofX. How many gallons of ice cream should the store
have on hand each of these days, so that the probability of exhausting its supply
on a particular day is 0.05?
1.7.19.Find the 25th percentile of the distribution having pdff(x)=|x|/ 4 ,where
− 2 <x<2 and zero elsewhere.