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,