probability and random variables exams solution

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Math 431 An Introduction to Probability Final Exam — Solutions

1.

A continuous random variable X has cdf F (x) =

    a

x

   b

2

for x ≤ 0, for 0 < x < 1, for x ≥ 1.

(a) Determine the constants a and b. (b) Find the pdf of X. Be sure to give a formula for fX (x) that is valid for all x. (c) Calculate the expected value of X. (d) Calculate the standard deviation of X. Answer: (a) We must have a = limx→−∞ = 0 and b = limx→+∞ = 1, since F is a cdf. (b) For all x 6= 0 or 1, F is differentiable at x, so 0

f (x) = F (x) =

(

2x if 0 < x < 1, 0 otherwise.

(One could also use any f that agrees with this definition for all x 6= 0 or 1.) (c) E(X) =

R∞

−∞

x · f (x) dx =

R1 0

x · 2x dx = 32 .

(d) E(X 2 ) = 01 xq2 · 2x dx = 12 , Var(X) = E(X 2 ) − [E(X)]2 = q 1 and σ(X) = Var(X) = 18 = 3√1 2 ≈ .2357. R

1 2

− ( 23 )2 =

1 18

≈ 0.0556,


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