Week 8 Discussion

Suppose again that Z=X+Y. Find \(f_Z\) if

• \(f_X(x) = \frac{1}{\sqrt{2\pi}σ_1} * e^{-(x-μ_1)^2 /2{σ_1}^2}\)
• \(f_Y(y) = \frac{1}{\sqrt{2\pi}σ_2} * e^{-(x-μ_2)^2 /2{σ_2}^2}\)

Thus we know; \[ f_Z(z) = \frac{1}{\sqrt{2\pi}\sqrt{σ_1^2 + σ_2^2}} * e^{-(z-μ_1 -μ_2)^2 /2(σ_1^2 + σ_2 ^2) )} \]

The above is true if they’re independent. Var(X+Y) = Var(X)+Var(Y) + 2Cov(X,Y), where Cov(X,Y) = 0

If independence fails, then Cov(X,Y) is not 0. This follows the Cauchy-Schwartz Inequality \[ Cov(X,Y) = σ_{X,Y} \leq \sqrt{σ_1^2 * σ_2^2} \] Then, Z follows a normal distribution with the following parameters; \[ Z \sim N(μ_1 + μ_2, σ_1^2 + σ_2^2 +2Cov(X,Y)) \]