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[answered] Math 170A - Homework 5 - Due 22 November 2016 1. Items prod


i just need 3 problems of this homework pdf.:

1. problem 4(i)

2.problem 5?

3.problem 6


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the related lecture notes are on pdf: Notes.28-39.pdf.


Math 170A - Homework 5 - Due 22 November 2016

 

1. Items produced by a machine have a defective rate of 0.5%. Estimate the probability that

 

the first defective item is between the 100th and the 300th item inclusive.

 

2. If X ? Exp(2) and Y ? Unif[0, 3] are independent, what is the joint density of (X, Y )?

 

3. (i) If X is an absolutely continuous random variable with density f (t) = ce?2t I(?1 ?

 

t ? 1), what is c?

 

(ii) If X and Y are jointly absolutely continuous random variable with density f (x, y) =

 

c|x|I(x2 + y 2 ? 1), what is c? 4.  

 

1

 

, what is P(1 ? X ? 2)?

 

(i) If X ? Exp(?) and ? ? Geom

 

2

 



 



 

1

 

(ii) If X ? Bern(p) and p ? Unif 0, , what is P(X = 1)?

 

2

 

(iii) If X ? Unif[0, 2?] and ? is an absolutely continuous random variable with density

 

f (t) = 3t2 I(0 ? t ? 1), what is P(X ? 1)? 5. Suppose that X and Y are jointly absolutely continuous random variables with density

 

(

 

3 2

 

2

 

2 (x + y ), if x, y ? [0, 1] .

 

f (x, y) =

 

0,

 

Otherwise

 

1

 

(i) What is the conditional density of X given y = ?

 

4 

 



 

1 1

 

(ii) Find P X ? y =

 

.

 

2

 

4

 

6. Suppose that Y ? Exp(X) and X ? Unif[1, 4].

 

(i) Find the joint density of (X, Y ).

 

(ii) Find the conditional density of X given y = 2. You may leave an integral in your

 

answer.

 

(iii) Find P(2 ? X ? 3|y = 2). You may leave two integrals in your answer. 1

 


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