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Chapter 7: Q. 7.25 (page 354)

Let X1,X2,…be a sequence of independent and identically distributed continuous random variables. Let N≥2be such that

X1≥X2≥⋯≥XN-1<XN

That is, Nis the point at which the sequence stops decreasing. Show that E[N]=e.

Hint: First find P{N≥n}.

Short Answer

Expert verified

PX1≥X2≥…≥XN-1<Xn=1-1(N-1)!N-1N=1(N-2)!N

We have proved thatE[N]=e.

Step by step solution

01

Given Information

X1,X2,…be independent and identically distributed continuous random variables.

Let N≥2be such that X1≥X2≥…≥XN-1<XN

02

Calculation

PX1≥X2≥…≥XN-1<XN

=PX1≥X2≥…≥XN-1PXN>XN-1∣X1≥X2≥…≥XN-1

We known X1≥X2≥…≥XN-1because (N-1)!

we have PX1≥X2≥…≥XN-1=1(N-1)!

PXN>XN-1∣X1≥X2≥…≥XN-1=1-1N=N-1N

PX1≥X2≥…≥XN-1<Xn=1-1(N-1)!N-1N

=1(N-2)!N

03

Final Answer

E[N]=∑N=2∞N1(N-2)!N

=∑N=2∞1(N-2)!

=∑N=0∞1N!

=e.

Therefore,E[N]=e.

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Most popular questions from this chapter

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a∣]is minimized

when ais equal to the median of F.

Hint: Write

E[IX-al]=|x-a|f(x)dx

Now break up the integral into the regions where x<a

and where x>a, and differentiate.

The joint density of X and Y is given by

f(x,y)=12πe-ye-(x-y)2/20<y<∞,

-∞<x<∞

(a) Compute the joint moment generating function of X and Y.

(b) Compute the individual moment generating functions.

There are n items in a box labeled H and m in a box labeled T. A coin that comes up heads with probability p and tails with probability 1 − p is flipped. Each time it comes up heads, an item is removed from the H box, and each time it comes up tails, an item is removed from the T box. (If a box is empty and its outcome occurs, then no items are removed.) Find the expected number of coin flips needed for both boxes to become empty. Hint: Condition on the number of heads in the first n + m flips.

Let Xbe a random variable having finite expectation μand variance σ2, and let g(*)be a twice differentiable function. Show that

E[g(X)]≈g(μ)+g''(μ)2σ2

Hint: Expand g(·)in a Taylor series about μ. Use the first

three terms and ignore the remainder.

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(a) Find E[X]and E[Y].
(b) Find Cov(X,Y).
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