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Chapter 5: Q.5.6 (page 212)

5.6. ComputeE[X]if Xhas a density function given by

(a)f(x)=14xe-x2x>00otherwise;

(b)f(x)=c(1鈭�x2)-1<x<10otherwise;

(c)f(x)=5x2x>50x鈮�5.

Short Answer

Expert verified

Thus the answer of given question is

(a),E(X)=4

(b),E(X)=0

(c),E(X)=鈭�

Step by step solution

01

Given Information.

f(x)=14xe-x2x>00otherwise

02

Explanation.

E(x)=鈭�-鈭�鈭�x.14xe-x2dx

=14鈭�0鈭�x9e-x2dx

x2=4,x=2y

E(X)=鈭�0鈭�2.y2e-ydy=2鈭�0鈭�y3-1e-1dy

=2脳2=4

03

Explanation.

f(x)=c(1-x2)鈭�1<x<10otherwise

鈭�-鈭�鈭�(1-x2)dx=1,鈭�-鈭�鈭�c(1-x2)dx=1

c.43=1鈬�c=34

E(X)=鈭�-鈭�鈭�x.34(1-x2)dx=34鈭�-11(x-34)dx

=34x22-x441-1

=34脳0=0

04

Explanation.

E(X)=鈭�-鈭�鈭�5x0dx=鈭�50xdx=5

=5ux15鈭�

=鈭�

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Most popular questions from this chapter

An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with 渭=4and蟽2=4, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters(6,9). A point is randomly chosen on the image and has a reading of5. If the fraction of the image that is black is伪, for what value of伪 would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?

The standard deviation of X, denoted SD(X), is given by

SD(X)=Var(X)

Find SD(aX+b) if X has variance 蟽2.

If X is an exponential random variable with parameter 位, and c > 0, show that cX is exponential with parameter 位/c

Show thatE[Y]=鈭�0鈭�PY>ydy-鈭�0鈭�PY<-ydy.

Hint: Show that 鈭�0鈭�PY<-ydy=-鈭�-鈭�0xfy(x)dx

鈭�0鈭�PY>ydy=鈭�0鈭�xfy(x)dx

The probability density function of X, the lifetime of a certain type of electronic device (measured in hours), is given by

f(x)=10x2x>100x鈮�10

(a)FindP{X>20}

localid="1646589462481" (b)What is the cumulative distribution function of localid="1646589521172" X?

localid="1646589534997" (c)) What is the probability that of6such types of devices, at least localid="1646589580632" 3will function for at least localid="1646589593287" 15hours? What assumptions are you making?
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