44 Probability and DistributionsEXERCISES1.5.1.Let a card be selected from an ordinary deck of playing cards. The outcome
cis one of these 52 cards. LetX(c)=4ifcis an ace, letX(c)=3ifcis a king,
letX(c)=2ifcis a queen, letX(c)=1ifcis a jack, and letX(c)=0otherwise.
Suppose thatPassigns a probability of 521 to each outcomec. Describe the induced
probabilityPX(D)onthespaceD={ 0 , 1 , 2 , 3 , 4 }of the random variableX.
1.5.2.For each of the following, find the constantcso thatp(x) satisfies the con-
dition of being a pmf of one random variableX.(a)p(x)=c(^23 )x,x=1, 2 , 3 ,..., zero elsewhere.(b)p(x)=cx,x=1, 2 , 3 , 4 , 5 ,6, zero elsewhere.1.5.3.LetpX(x)=x/15,x=1, 2 , 3 , 4 ,5, zero elsewhere, be the pmf ofX.Find
P(X= 1 or 2),P(^12 <X<^52 ), andP(1≤X≤2).
1.5.4.LetpX(x) be the pmf of a random variableX.FindthecdfF(x)ofXand
sketch its graph along with that ofpX(x)if:
(a)pX(x)=1,x= 0, zero elsewhere.(b)pX(x)=^13 ,x=− 1 , 0 ,1, zero elsewhere.(c)pX(x)=x/ 15 ,x=1, 2 , 3 , 4 ,5, zero elsewhere.1.5.5.Let us select five cards at random and without replacement from an ordinary
deck of playing cards.(a)Find the pmf ofX, the number of hearts in the five cards.(b)DetermineP(X≤1).∫1.5.6.Let the probability set function of the random variableXbePX(D)=
Df(x)dx,wheref(x)=2x/9, forx∈D={x:0<x<^3 }. Define the events
D 1 ={x:0<x< 1 }andD 2 ={x:2<x< 3 }. ComputePX(D 1 ),PX(D 2 ), and
PX(D 1 ∪D 2 ).
1.5.7.Let the space of the random variableX beD={x:0<x< 1 }.If
D 1 ={x:0<x<^12 }andD 2 ={x:^12 ≤x< 1 }, findPX(D 2 )ifPX(D 1 )=^14.1.5.8.Suppose the random variableXhas the cdfF(x)=⎧
⎨
⎩0 x<− 1
x+2
4 −^1 ≤x<^1
11 ≤x.Write an R function to sketch the graph ofF(x). Use your graph to obtain the
probabilities: (a)P(−^12 <X≤^12 ); (b)P(X= 0); (c)P(X= 1); (d)P(2<X≤3).