Let [math]X[/math] be a random variable with range [math][-1,1][/math] and let [math]f_X(x)[/math] be the density function of [math]X[/math]. Find [math]\mu(X)[/math] and [math]\sigma^2(X)[/math] if, for [math]|x| \lt 1[/math],

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

Let [math]X[/math] be a random variable with range [math][-1,1][/math] and [math]f_X[/math] its density function. Find [math]\mu(X)[/math] and [math]\sigma^2(X)[/math] if, for [math]|x| \gt 1[/math], [math]f_X(x) = 0[/math], and for [math]|x| \lt 1[/math],

• [math]f_X(x) = (3/4)(1 - x^2)[/math].
• [math]f_X(x) = (\pi/4)\cos(\pi x/2)[/math].
• [math]f_X(x) = (x + 1)/2[/math].
• [math]f_X(x) = (3/8)(x + 1)^2[/math].

The lifetime, measure in hours, of the ACME super light bulb is a random variable [math]T[/math] with density function [math]f_T(t) = \lambda^2 t e^{-\lambda t}[/math], where [math]\lambda = .05[/math]. What is the expected lifetime of this light bulb? What is its variance?

Let [math]X[/math] be a random variable with range [math][-1,1][/math] and density function [math]f_X(x) = ax + b[/math] if [math]|x| \lt 1[/math].

• Show that if [math]\int_{-1}^{+1} f_X(x)\, dx = 1[/math], then [math]b = 1/2[/math].
• Show that if [math]f_X(x) \geq 0[/math], then [math]-1/2 \leq a \leq 1/2[/math].
• Show that [math]\mu = (2/3)a[/math], and hence that [math]-1/3 \leq \mu \leq 1/3[/math].
• Show that [math]\sigma^2(X) = (2/3)b - (4/9)a^2 = 1/3 - (4/9)a^2[/math].

Let [math]X[/math] be a random variable with range [math][-1,1][/math] and density function [math]f_X(x) = ax^2 + bx + c[/math] if [math]|x| \lt 1[/math] and 0 otherwise.

• Show that [math]2a/3 + 2c = 1[/math] (see Exercise).
• Show that [math]2b/3 = \mu(X)[/math].
• Show that [math]2a/5 + 2c/3 = \sigma^2(X)[/math].
• Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/15[/math], and sketch the graph of [math]f_X[/math].
• Find [math]a[/math], [math]b[/math], and [math]c[/math] if [math]\mu(X) = 0[/math], [math]\sigma^2(X) = 1/2[/math], and sketch the graph of [math]f_X[/math].

Let [math]T[/math] be a random variable with range [math][0,\infty][/math] and [math]f_T[/math] its density function. Find [math]\mu(T)[/math] and [math]\sigma^2(T)[/math] if, for [math]t \lt 0[/math], [math]f_T(t) = 0[/math], and for [math]t \gt 0[/math],

• [math]f_T(t) = 3e^{-3t}[/math].
• [math]f_T(t) = 9te^{-3t}[/math].
• [math]f_T(t) = 3/(1 + t)^4[/math].

Let [math]X[/math] be a random variable with density function [math]f_X[/math]. Show, using elementary calculus, that the function

takes its minimum value when [math]a = \mu(X)[/math], and in that case [math]\phi(a) = \sigma^2(X)[/math].

Let [math]X[/math] be a random variable with mean [math]\mu[/math] and variance [math]\sigma^2[/math]. Let [math]Y = aX^2 + bX + c[/math]. Find the expected value of [math]Y[/math].

Let [math]X[/math], [math]Y[/math], and [math]Z[/math] be independent random variables, each with mean [math]\mu[/math] and variance [math]\sigma^2[/math].

• Find the expected value and variance of [math]S = X + Y + Z[/math].
• Find the expected value and variance of [math]A = (1/3)(X + Y + Z)[/math].
• Find the expected value of [math]S^2[/math] and [math]A^2[/math].

Let [math]X[/math] and [math]Y[/math] be independent random variables with uniform density functions on [math][0,1][/math]. Find

• [math]E(|X - Y|)[/math].
• [math]E(\max(X,Y))[/math].
• [math]E(\min(X,Y))[/math].
• [math]E(X^2 + Y^2)[/math].
• [math]E((X + Y)^2)[/math].