The joint probability density for [math]X[/math] and [math]Y[/math] is

[[math]] f(x,y) = \begin{cases} 2e^{-(x + 2y)}, \,\, x \gt 0, y \gt 0\\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

Calculate the variance of [math]Y[/math] given that [math]X \gt 3 [/math] and [math]Y \gt 3 [/math].

• 0.25
• 0.50
• 1.00
• 3.25
• 3.50

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Step 1: Identify Independence and Marginal Distributions

The joint probability density function for [math]X[/math] and [math]Y[/math] is given by:

[[math]]f(x,y) = \begin{cases} 2e^{-(x + 2y)}, \,\, x \gt 0, y \gt 0\\ 0, \, \textrm{Otherwise.}\end{cases}[[/math]]

We can factorize [math]f(x,y)[/math] into a product of two functions, one depending only on [math]x[/math] and the other only on [math]y[/math]:

[[math]]f(x,y) = e^{-x} \cdot 2e^{-2y}[[/math]]

This factorization holds for the specified domain [math]x \gt 0[/math] and [math]y \gt 0[/math]. Since the joint PDF can be expressed as a product of two functions, each solely dependent on one variable, and the support is a rectangular region, [math]X[/math] and [math]Y[/math] are independent random variables. The marginal probability density functions are:

• [math]f_X(x) = e^{-x}[/math] for [math]x \gt 0[/math], which indicates that [math]X[/math] follows an exponential distribution with rate parameter [math]\lambda_X = 1[/math].
• [math]f_Y(y) = 2e^{-2y}[/math] for [math]y \gt 0[/math], which indicates that [math]Y[/math] follows an exponential distribution with rate parameter [math]\lambda_Y = 2[/math].

Step 2: Address the Conditional Information

We are asked to calculate [math]Var(Y | X \gt 3, Y \gt 3)[/math]. Impact of X > 3: Since [math]X[/math] and [math]Y[/math] are independent (as established in Step 1), any information about [math]X[/math] (such as [math]X \gt 3[/math]) does not affect the distribution or moments of [math]Y[/math]. Therefore, the condition [math]X \gt 3[/math] can be ignored. The problem simplifies to finding [math]Var(Y | Y \gt 3)[/math]. Impact of Y > 3: The exponential distribution possesses the memoryless property. This property states that for an exponential random variable [math]Y[/math] with rate parameter [math]\lambda[/math], the conditional distribution of [math]Y - t[/math] given [math]Y \gt t[/math] is the same as the unconditional distribution of [math]Y[/math]. In our case, with [math]t=3[/math], the distribution of [math]Y | Y \gt 3[/math] is equivalent to the distribution of [math]3 + Y'[/math], where [math]Y'[/math] is an exponential random variable with the same rate parameter [math]\lambda_Y = 2[/math] as [math]Y[/math]. We want to calculate [math]Var(Y | Y \gt 3)[/math]. Using the relationship derived from the memoryless property:

[[math]]Var(Y | Y \gt 3) = Var(3 + Y')[[/math]]

A fundamental property of variance is that adding a constant to a random variable does not change its variance: [math]Var(c + Z) = Var(Z)[/math] for any constant [math]c[/math] and random variable [math]Z[/math]. Applying this property:

[[math]]Var(3 + Y') = Var(Y')[[/math]]

Since [math]Y'[/math] has the same distribution as [math]Y[/math] (i.e., [math]Y' \sim \text{Exp}(\lambda_Y = 2)[/math]), it follows that [math]Var(Y') = Var(Y)[/math]. Therefore, we conclude that [math]Var(Y | X \gt 3, Y \gt 3) = Var(Y)[/math].

Step 3: Calculate the Variance of Y

From Step 1, we established that [math]Y[/math] follows an exponential distribution with rate parameter [math]\lambda_Y = 2[/math]. For an exponential distribution with parameter [math]\lambda[/math], the variance is given by the formula [math]1/\lambda^2[/math]. Using this formula for [math]Y[/math]:

[[math]]Var(Y) = \frac{1}{\lambda_Y^2} = \frac{1}{2^2} = \frac{1}{4} = 0.25[[/math]]

Thus, the variance of [math]Y[/math] given that [math]X \gt 3[/math] and [math]Y \gt 3[/math] is [math]0.25[/math].

Key Insights

• Independence from Joint PDF: If a joint probability density function [math]f(x,y)[/math] can be factored into [math]f_X(x)f_Y(y)[/math] over a rectangular support, then the random variables [math]X[/math] and [math]Y[/math] are independent.
• Conditional Probabilities with Independence: When random variables are independent, a condition on one variable does not affect the distribution or moments of the other variable.
• Memoryless Property of Exponential Distribution: The exponential distribution is memoryless. This means [math]Var(Y | Y \gt t) = Var(Y)[/math] for an exponential random variable [math]Y[/math] and any [math]t \gt 0[/math]. More generally, the distribution of [math]Y - t[/math] given [math]Y \gt t[/math] is the same as the unconditional distribution of [math]Y[/math].
• Variance of a Constant Shift: Adding or subtracting a constant to a random variable does not change its variance. That is, [math]Var(Y + c) = Var(Y)[/math] for any constant [math]c[/math].
• Moments of Exponential Distribution: For an exponential distribution with rate parameter [math]\lambda[/math], the mean is [math]1/\lambda[/math] and the variance is [math]1/\lambda^2[/math].

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