Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

166 Some Special Distributions

3.1.21.LetX 1 andX 2 have a trinomial distribution. Differentiate the moment-

generating function to show that their covariance is−np 1 p 2.

3.1.22.If a fair coin is tossed at random five independent times, find the conditional
probability of five heads given that there are at least four heads.

3.1.23.Let an unbiased die be cast at random seven independent times. Compute
the conditional probability that each side appears at least once given that side 1
appears exactly twice.

3.1.24.Compute the measures of skewness and kurtosis of the binomial distribution
b(n, p).

3.1.25.Let

p(x 1 ,x 2 )=

(

x 1

x 2

)(

1

2

)x 1 (

x 1

15

)

,

x 2 =0, 1 ,...,x 1

x 1 =1, 2 , 3 , 4 , 5 ,

zero elsewhere, be the joint pmf ofX 1 andX 2. Determine

(a)E(X 2 ).

(b)u(x 1 )=E(X 2 |x 1 ).

(c)E[u(X 1 )].

Compare the answers of parts (a) and (c).
Hint: Note thatE(X 2 )=

∑ 5

x 1 =1

∑x 1

x 2 =0x^2 p(x^1 ,x^2 ).

3.1.26.Three fair dice are cast. In 10 independent casts, letXbe the number of
times all three faces are alike and letY be the number of times only two faces are
alike. Find the joint pmf ofXandYand computeE(6XY).

3.1.27.LetXhave a geometric distribution. Show that

P(X≥k+j|X≥k)=P(X≥j), (3.1.9)

wherekandjare nonnegative integers. Note that we sometimes say in this situation
thatXismemoryless.

3.1.28.LetXequal the number of independent tosses of a fair coin that are required
to observe heads on consecutive tosses. Letunequal thenth Fibonacci number,
whereu 1 =u 2 =1andun=un− 1 +un− 2 ,n=3, 4 , 5 ,....

(a)Show that the pmf ofXis

p(x)=

ux− 1

2 x

,x=2, 3 , 4 ,....

(b)Use the fact that

un=

1

√

5

[(

1+

√

5

2

)n

−

(

1 −

√

5

2

)n]

to show that

∑∞

x=2p(x)=1.