Probability: Midterm 2 Solutions
Fall, 2009
Instructor: W. D. Gillam
(1) (3 points) Give the definitions of the beta and gamma functions, the relationship
between them, and the functional equation for the gamma function. (No proofs
required.)
(2) (4 points) Give the density function for a random variable Xwith normal distri-
bution with expected value µand variance σ2. Express P(a≤X≤b) as a definite
integral of the standard normal density function.
Solution. (X−µ)/√σ2has a standard normal distribution, so
P(a≤X≤b) = P(a−µ
√σ2≤X−µ
√σ2≤b−µ
√σ2)
=1
√2πZb−µ
√σ2
a−µ
√σ2
e−x2/2dx.
(3) (6 points) Give the density functions and expected values for the gamma, beta,
and Student distributions. Indicate the choices of the parameters α, β for which
the gamma density function specializes to the chi squared density function with
ndegrees of freedom. Explain how the chi squared distribution and Student’s
distribution arise from the normal distribution.
(4) (4 points) Suppose Xhas a binomial distribution based on ntrials with success
probability p. Calculate the moment generating function gX(t) of X.Hint: You
can use multiplicativity of MGFs under sums of independent random variables to
reduce to the case n= 1.
Solution. When n= 1, this is gX(t) = E(etX ) = pet·1+qet·0=pet+q, so in
general it is gX(t) = (pet+q)n.
(5) (4 points) Suppose Xhas a beta distribution with parameters α=β= 1/2.
Find the cumulative distribution function FX(x) of X.
Solution. The density function for Xis given by
f(y) = Γ(1)
Γ(1/2)2y−1/2(1 −y)−1/2
=1
π
1
√y√1−y
