Exercise 5.54 Let X be a random variable with the exponential distribution, parameter . Find the density function of (a) A = 2X + 5,

Exercise 5.54 Let X be a random variable with the exponential distribution, parameter λ. Find the density function of

(a) A = 2X + 5,

(b) B = eX ,

(c) C = (1 + X)−1,

(d) D = (1 + X)−2.

Exercise 5.55 Show that if X has the normal distribution with parameters 0 and 1, then Y = X2 has the

χ2 distribution with one degree of freedom Example 5.65 If X has the normal distribution with parameters μ = 0 and σ2 = 1, then E(X) = Z ∞
−∞
x 1 √2π
e−1 2 x2 dx = 0, by symmetry properties of the integrand. Hence var(X) = Z ∞
−∞
x2 1 √2π
e−1 2 x2 dx = 1.
Similar integrations show that the normal distribution with parameters μ and σ2 has mean μ
and variance σ2, as we may have expected. △
Example 5.66 If X has the Cauchy distribution, then E(X) = Z ∞
−∞
x 1 π(1 + x2)
dx, so long as this integral exists. It does not exist, since Z N −M x 1 π(1 + x2)
dx = 
1 2π
log(1 + x2)
N −M = 1 2π
log 1 + N2