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Chapter 8: Q. 8.13 (page 393)

Show that ifE[X]<0androle="math" localid="1649871241073" 胃鈮�0is such thatEe胃X=1, then胃>0.

Short Answer

Expert verified

Consider the functionf(y)=-ln(y),y>0. Since this function is convex, if we letY=e胃X, using Jensen's inequalityE[f(Y)]鈮�f(E[Y]),

Step by step solution

01

Given Information.

E[X]<0and 胃鈮�0is such thatEe胃X=1, then胃>0.

02

Explanation.

Assume thatE[X]<0and 胃鈮�0such thatEe胃X=1, whereby 胃is a scalar. We want to show that胃>0. To do this, we will use Jensen's inequality. It says that for a convex function fand an arbitrary random variable Yis

E[f(Y)]鈮�f(E[Y])

Consider the functionf(y)=-ln(y),y>0. Since this function is convex, if we letY=e胃X, using Jensen's inequality, we get:

E-lne胃X鈮�-lnEe胃X

Sincelneu=u, for eachu, and using the informationEe胃X=1, we get:

E[-胃X]鈮�-ln(1)=0

if properties of expectation

-胃E[X]鈮�0

鈬�it is givenE[X]<0

-胃<0

-胃<0

胃>0.

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Most popular questions from this chapter

Civil engineers believe that W, the amount of weight (in units of 1000pounds) that a certain span of a bridge can withstand without structural damage resulting, is normally distributed with a mean of 400and standard deviation of40. Suppose that the weight (again, in units of 1000pounds) of a car is a random variable with a mean of 3and standard deviation.3. Approximately how many cars would have to be on the bridge span for the probability of structural damage to exceed.1?

Let Xbe a Poisson random variable with a mean of20.

(a)Use the Markov inequality to obtain an upper boundp=P(X鈮�26).

(b)Use the one-sided Chebyshev inequality to obtain an upper boundp.

(c)Use the Chernoff bound to obtain an upper boundp.

(d)Approximate pby making use of the central limit theorem.

(e)Determine pby running an appropriate program.

The strong law of large numbers states that with probability 1, the successive arithmetic averages of a sequence of independent and identically distributed random variables converge to their common mean . What do the successive geometric averages converge to? That is, what is

limn鈫�鈭�鈭�i=1nXi1/n

It Xhas a variance蟽2, then 蟽, the positive square root of the variance, is called the standard deviation. It Xhas to mean 渭and standard deviation蟽, to show thatP{|X-渭|鈮�k蟽}鈮�1k2.

ItXis a Poisson random variable with a mean100, thenPX>120is approximately

(a).02,

(b).5or

(c).3?
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