1.7. Continuous Random Variables 49(b)Show that∑∞
x=1p(x)=1.
(c)DetermineP(X=1, 3 , 5 , 7 ,...).(d)Find the cdfF(x)=P(X≤x).1.6.4.Cast a die two independent times and letXequal the absolute value of the
difference of the two resulting values (the numbers on the up sides). Find the pmf
ofX.Hint:It is not necessary to find a formula for the pmf.
1.6.5.For the random variableXdefined in Example 1.6.2:
(a)Write an R function that returns the pmf. Note that in R,choose(m,k)
computes(m
k)
.(b)Write an R function that returns the the graph of the cdf.
1.6.6.For the random variableXdefined in Example 1.6.1, graph the cdf ofX.
1.6.7.LetXhave a pmfp(x)=^13 ,x=1, 2 ,3, zero elsewhere. Find the pmf of
Y=2X+1.
1.6.8.LetXhave the pmfp(x)=(^12 )x,x=1, 2 , 3 ,..., zero elsewhere. Find the
pmf ofY=X^3.1.6.9.LetXhave the pmfp(x)=1/3,x=− 1 , 0 ,1. Find the pmf ofY=X^2.
1.6.10.LetXhave the pmfp(x)=(
1
2)|x|
,x=− 1 ,− 2 ,− 3 ,....Find the pmf ofY=X^4.
1.6.11.Show that the function given in expression (1.6.6) is a pmf.1.7 ContinuousRandomVariables
In the last section, we discussed discrete random variables. Another class of random
variables important in statistical applications is the class of continuous random
variables, which we define next.
Definition 1.7.1(Continuous Random Variables).We say a random variable is a
continuous random variableif its cumulative distribution functionFX(x)is a
continuous function for allx∈R.
Recall from Theorem 1.5.3 thatP(X=x)=FX(x)−FX(x−), for any random
variableX. Hence, for a continuous random variableX, there are no points of
discrete mass; i.e., ifX is continuous, thenP(X =x) = 0 for allx∈R.Most
continuous random variables areabsolutely continuous;thatis,
FX(x)=∫x−∞fX(t)dt, (1.7.1)