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Chapter 2: Problem 1

Let \(X_{1}\) and \(X_{2}\) have the joint pdf \(f\left(x_{1}, x_{2}\right)=x_{1}+x_{2}, 0

Short Answer

Expert verified
The conditional mean and variance require multi-step calculations involving integrals. Once the marginal pdf of \(X_{1}\) is computed, it is used to find the conditional pdf of \(X_{2}\) given \(X_{1}=x_{1}\). This conditional pdf is used to calculate the conditional mean and variance of \(X_{2}\) given \(X_{1}=x_{1}\).

Step by step solution

01

Compute the marginal pdf of \(X_{1}\)

The marginal pdf of \(X_{1}\), denoted \(g(x_{1})\), can be computed by integrating the joint pdf over all possible values of \(X_{2}\), from 0 to 1: \[ g(x_{1}) = \int_0^1 f(x_{1}, x_{2}) dx_{2} = \int_0^1 (x_{1} + x_{2}) dx_{2} .\] Compute the integral to find \(g(x_{1})\).
02

Compute the conditional pdf of \(X_{2}\) given \(X_{1}=x_{1}\)

The conditional pdf of \(X_{2}\) given \(X_{1}=x_{1}\), denoted \(h(x_{2}|X_{1}=x_{1})\), is computed as the ratio of the joint pdf to the marginal pdf of \(X_{1}\): \[h(x_{2}|X_{1}=x_{1}) = \frac{f(x_{1}, x_{2})}{g(x_{1})}. \] Substitute in the expressions for \(f(x_{1}, x_{2})\) and \(g(x_{1})\) and simplify to find \(h(x_{2}|X_{1}=x_{1})\).
03

Compute the conditional mean of \(X_{2}\) given \(X_{1}=x_{1}\)

The conditional mean, denoted \(\mathbb{E}[X_{2}|X_{1}=x_{1}]\), is computed as \[\mathbb{E}[X_{2}|X_{1}=x_{1}] = \int_0^1 x_{2} h(x_{2}|X_{1}=x_{1}) dx_{2}.\] Substitute the expression for \(h(x_{2}|X_{1}=x_{1})\) and compute the integral.
04

Compute the conditional variance of \(X_{2}\) given \(X_{1}=x_{1}\)

The conditional variance, denoted \(\operatorname{Var}(X_{2}|X_{1}=x_{1})\), is computed as \[\operatorname{Var}(X_{2}|X_{1}=x_{1}) = \mathbb{E}[X_{2}^2|X_{1}=x_{1}] - (\mathbb{E}[X_{2}|X_{1}=x_{1}])^2.\] The term \(\mathbb{E}[X_{2}^2|X_{1}=x_{1}]\) can be computed similarly to the conditional mean in step 3, except you integrate \(x_{2}^2\) against \(h(x_{2}|X_{1}=x_{1})\). The term \((\mathbb{E}[X_{2}|X_{1}=x_{1}])^2\) is found by squaring the result from step 3. Subtract the two results to get the conditional variance.

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