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[answered] Mathematics 170A - Homework 8: Due Thursday, Nov. 17, 2016.


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Mathematics 170A - Homework 8: Due

 

Thursday, Nov. 17, 2016. Problem 1 on Page 184, problems 5, 6, 11 on Pages 186-188. A. For Problem E in Homework 6

 

(a) Compute the marginal PMFs of D and of M.

 

(b) Are D and M independent? Explain. B. The random variables X1 , X2 , ..., Xn are independent, and they all have

 

the same mean, but have different variance ?i2 . We would like to

 

Ptry to

 

estimate the common mean by the value of random variable S = i ai Xi ,

 

for some choice of constants ai .

 

(a) Under what condition on the a0i s is E[S] = E[X1 ], that is, equalling to

 

the common mean?

 

(b) Extra Credit: Among all choices of the a0i s that satisfy E[S] = E[X1 ],

 

find the one that minimizes Var(S). C. Suppose that X has CDF 0 if x < 0, x if 0 ? x < 1,

 

F (x) = x3 if 1 ? x < 2, 2

 

1 if x ? 2.

 

1 (1) (a) Find P( 12 ? X ? 32 ).

 

(b) Find P( 12 ? X ? 1).

 

(c) P( 21 ? X < 1).

 

(d) P(1 ? X ? 32 ).

 

(e) P(1 < X < 2).

 

D. Suppose that X is uniform on [0, 2]. Find the CDF of Y = min(X, a).

 

E. Suppose that X is exponentially distributed. If P(X ? 0.02) = 0.4, find

 

a number x so that P(X ? x) = 0.8.

 

F. A point (X, Y ) is chosen uniformly from the unit square [0, 1]?[0, 1]. Find

 

the CDF of the random variable Z = X + Y .

 

G. Suppose that X has a symmetric density f (x) that is f (?x) = f (x). Find

 

f if X 2 is exponentially distributed with parameter ?. 2

 


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