# Chapter 2 Stochastics Homework Solution

**2-8 Show that the binomial probabilities sum to 1.**

X ~ B (n, p)

**2-11 Consider the binomial distribution with n trails and probability**

**p of success on each trial. For what value of k is P(X=k)**

**maximized? This value is called the mode of the distribution.**

**(Hint: Consider the ratio of successive terms.)**

&

－(1)

－(2)

(1) (2)

if , k = p(n+1)-1 or p(n+1)

if , k = [ p(n+1) ]

**2-15 Two terms, A and B, play a series of games. If team A has**

**probability 0.4 of winning each game, is it to its advantage to**

**play the best three out of five games or the best four out of**

**seven? Assume the outcomes of successive games are**

**independent.**

(1) Three out of five games :

= 4.96 x 0.4

^{3}

(2) Four out of seven games :

“Three out of five games” is better!

**2-25 The probability of being dealt a royal straight flush (ace, king,**

**queen, jack, and ten of the same suit) in poker is about 1.3×10**

^{-8}.**Suppose that an avid poker player sees 100 hands a week, 52**

**weeks a year, for 20 years.**

**a. What is the probability that she never sees a royal straight**

**flush dealt?**

**b. What is the probability that she sees two royal straight flushes**

**dealt?**

100 x 52 x 20 = 104000

(a) ( 1.3 x 10

^{-8})

^{104000}= 0.9987

(b)

**2-28 Let p**

_{0}, p_{1}, …, p_{n}denote the probability mass function of the**binomial distribution with parameters n and p. Let q = 1-p.**

**Show that the binomial probabilities can be computed**

**recursively by p**

_{0}= q^{n}and

**k = 1, 2, …, n**

k = 1, 2, …, n

**2-29 Show that the Poisson probabilities p**

_{0}, p_{1}, …, p_{n}can be**computed recursively by p**

_{0}= exp(-λ) and

**k = 1, 2, …, n**

**2-32 For what value of k is the Poisson frequency function with**

**parameter λ maximized? (Hint: Consider the ratio of**

**consecutive terms.)**

&

&

if , k = λ or λ-1

if , k = [λ]

**2-39 The Cauchy cumulative distribution function is**

**-∞ ＜ x ＜∞**

**a Show that this is a cdf.**

**b Find the density function.**

**c Find x such that P (X ＞ x) = .1.**

-∞ ＜ x ＜∞

(a) (1)

(2)

(3) F(x) 右連續

By (1)(2)(3) F(x) is a c.d.f

(b)

(c)

**2-49 The gamma function is a generalized factorial function.**

**a Show that Γ(1) = 1.**

**b. Show that Γ(x+1) = xΓ(x). (Hint: Use integration by parts)**

**c. Conclude that Γ(n) = (n-1)! for n = 1, 2, 3, …**

**d. Use the fact that**

**to show that, if n is an odd integer,**

Gamma function

(a)

(b)

=

=

=

(c)

=

= …

=

=

(d) , n is an odd integer.

Claim:

Pf: (1) n = 1, .

(2) 設n = k (k is an odd integer.)

成立.

(3) n = k + 2.

=

By (1)(2)(3) 由數學歸納法知,原命題成立.

**2-51 Show that the normal density integrates to 1. (Hint: First**

**make a change of variables to reduce the integral to that for**

**the standard normal.) The problem is then to show that**

.

**Square both sides and reexpress the problem as**

**that of showing**

**=**

**Finally, write the product of integrals as a double integral and**

**change to the polar coordinates.**

claim:

i.e. claim:

Sol :

Let

=

=

=

=

=

=

**2-57**

**and , where a<0 , show that**

**Sol:**

X= ,

X= ,

=

=

_{}

**2-58**

**If U is uniform on [0,1] , find the density of .**

**Sol:**

**U ~uniform[0,1]**

, Y=

, Y=

(y)=2y ,

(y)=2y ,

**2-60**

**Find the density function of Y=, where .This is called the**

**lognormal density, since is normally distributed.**

**Sol:**

Z~N(μ,σ²)

Z~N(μ,σ²)

**Let Y= , Z= ,**

∴(y)=

∴(y)=

**=**

**2-63**

**Suppose that follows a uniform distribution on the interval [].Find the cdf and**

**density of tan.**

**Sol:**

**~U[]**

**= , θ[]**

**Y= tan , P(Yy)=P(tany)=P( ,**

**2-67**

**The Weibull cumulative distribution function is ,≧0,α>0,β>0**

**Find the density function****Show that is W follows a Weibull distribution , then X=follows an exponential distribution .****How could Weibull random variable be generated from a uniform random number generator ?**

**Sol:**

**, 0,α>0,β>0**

**=,0,α>0,β>0**

**, x0,α>0,β>0**

**X= ,**

**,**

**(c)**

**let z= 1-Z=**

**又 if Z~ uniform[0,1] then 1-Z~uniform[0,1]**

**∴we can firstly generate U ~uniform(0,1)**

**let W= then**

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