Problems 133
15.Is Problem 14 consistent with the results of Problems 12 and 13?
16.Suppose thatXandYare independent continuous random variables. Show that(a) P{X+Y≤a}=∫∞−∞FX(a−y)fY(y)dy(b)P{X≤Y}=∫∞−∞FX(y)fY(y)dywherefY is the density function ofY, andFXis the distribution function
ofX.
17.When a currentI(measured in amperes) flows through a resistanceR(measured
in ohms), the power generated (measured in watts) is given byW=I^2 R. Suppose
thatIandRare independent random variables with densitiesfI(x)= 6 x(1−x)0≤x≤ 1
fR(x)= 2 x 0 ≤x≤ 1Determine the density function ofW.
18.In Example 4.3b, determine the conditional probability mass function of the size
of a randomly chosen family containing 2 girls.
19.Compute the conditional density function ofXgivenY =yin(a)Problem 10
and(b)Problem 13.
20.Show thatXandYare independent if and only if
(a) PX/Y(x/y)=pX(x) in the discrete case
(b)fX/Y(x/y)=fX(x) in the continuous case
21.Compute the expected value of the random variable in Problem 1.
22.Compute the expected value of the random variable in Problem 3.
23.Each night different meteorologists give us the “probability” that it will rain the
next day. To judge how well these people predict, we will score each of them as
follows: If a meteorologist says that it will rain with probabilityp, then he or she
will receive a score of1 −(1−p)^2 if it does rain
1 −p^2 if it does not rainWe will then keep track of scores over a certain time span and conclude that
the meteorologist with the highest average score is the best predictor of weather.
Suppose now that a given meteorologist is aware of this and so wants to maximize
his or her expected score. If this individual truly believes that it will rain tomorrow