Let X be a uniform random variable on (0, k). a. Determine k such that V (X) = 3

Question

Let X be a uniform random variable on (0, k).

a. Determine k such that V (X) = 3E(X). (1 pt)

b. Using the k obtained in part a, write down the probability density function of X. (If you did not get part a, simply include k in your calculations and final answer.) (1 pt)

c. Using the k obtained in part a, find P( 1 X < 2). (If you did not get part a, simply include k in your calculations and final answer.) (1 pt)

d. Using the k obtained in part a, find the distribution function of X. (If you did not get part a, simply include k in your calculations and final answer.)

Explanation / Answer

V(X)=3E(X)

X =0 K

f(X) =1/2 1/2

E(X)=Sumx.f(x)

K*1

K

V(x)= E(x)2-(Ex)2

3E(x)=E(x)2-(Ex)2

E(x)2=K2/4

3E(X)=K2/4-K,We know E(X)=K

3K=K2/4-K

4K=K2/4

16K=K2

K=16

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