Let [math]X[/math] be a continuous random variable with values in [math][\,0,2][/math] and density [math]f_X[/math]. Find the moment generating function [math]g(t)[/math] for [math]X[/math] if

• [math]f_X(x) = 1/2[/math].
• [math]f_X(x) = (1/2)x[/math].
• [math]f_X(x) = 1 - (1/2)x[/math].
• [math]f_X(x) = |1 - x|[/math].
• [math]f_X(x) = (3/8)x^2[/math].

Hint: Use the integral definition, as in example and example.

For each of the densities in Exercise calculate the first and second moments, [math]\mu_1[/math] and [math]\mu_2[/math], directly from their definition and verify that [math]g(0) = 1[/math], [math]g'(0) = \mu_1[/math], and [math]g''(0) = \mu_2[/math].

Let [math]X[/math] be a continuous random variable with values in [math][\,0,\infty)[/math] and density [math]f_X[/math]. Find the moment generating functions for [math]X[/math] if

• [math]f_X(x) = 2e^{-2x}[/math].
• [math]f_X(x) = e^{-2x} + (1/2)e^{-x}[/math].
• [math]f_X(x) = 4xe^{-2x}[/math].
• [math]f_X(x) = \lambda(\lambda x)^{n - 1} e^{-\lambda x}/(n - 1)![/math].

For each of the densities in Exercise, calculate the first and second moments, [math]\mu_1[/math] and [math]\mu_2[/math], directly from their definition and verify that [math]g(0) = 1[/math], [math]g'(0) = \mu_1[/math], and [math]g''(0) = \mu_2[/math].

Find the characteristic function [math]k_X(\tau)[/math] for each of the random variables [math]X[/math] of Exercise.

Let [math]X[/math] be a continuous random variable whose characteristic function [math]k_X(\tau)[/math] is

Show directly that the density [math]f_X[/math] of [math]X[/math] is

Let [math]X[/math] be a continuous random variable with values in [math][\,0,1][/math], uniform density function [math]f_X(x) \equiv 1[/math] and moment generating function [math]g(t) = (e^t - 1)/t[/math]. Find in terms of [math]g(t)[/math] the moment generating function for

• [math]-X[/math].
• [math]1 + X[/math].
• [math]3X[/math].
• [math]aX + b[/math].

Let [math]X_1[/math], [math]X_2[/math],..., [math]X_n[/math] be an independent trials process with uniform density. Find the moment generating function for

• [math]X_1[/math].
• [math]S_2 = X_1 + X_2[/math].
• [math]S_n = X_1 + X_2 +\cdots+ X_n[/math].
• [math]A_n = S_n/n[/math].
• [math]S_n^* = (S_n - n\mu)/\sqrt{n\sigma^2}[/math].

Let [math]X_1[/math], [math]X_2[/math], ..., [math]X_n[/math] be an independent trials process with normal density of mean 1 and variance 2. Find the moment generating function for

• [math]X_1[/math].
• [math]S_2 = X_1 + X_2[/math].
• [math]S_n = X_1 + X_2 +\cdots+ X_n[/math].
• [math]A_n = S_n/n[/math].
• [math]S_n^* = (S_n - n\mu)/\sqrt{n\sigma^2}[/math].

Let [math]X_1[/math], [math]X_2[/math], ..., [math]X_n[/math] be an independent trials process with density

• Find the mean and variance of [math]f(x)[/math].
• Find the moment generating function for [math]X_1[/math], [math]S_n[/math], [math]A_n[/math], and [math]S_n^*[/math].
• What can you say about the moment generating function of [math]S_n^*[/math] as [math]n \to \infty[/math]?
• What can you say about the moment generating function of [math]A_n[/math] as [math]n \to \infty[/math]?