Let f1(y) and f2(y) be density functions and let a be a

QUESTION:

Let \(f_{1}(y) \text { and } f_{2}(y)\) be density functions and let 𝑎 be a constant such that \(0 \leq a \leq 1\). Consider the function \(f(y)=a f_{1}(y)+(1-a) f_{2}(y)\).

a Show that \(f(y)\) is a density function. Such a density function is often referred to as a

mixture of two density functions.

b Suppose that \(Y_{1}\) is a random variable with density function \(f_{1}(y)\) and that \(E\left(Y_{1}\right)=\mu_{1}\) and \(\operatorname{Var}\left(Y_{1}\right)=\sigma_{1}^{2}\); and similarly suppose that \(Y_{2}\) is a random variable with density function \(f_{2}(y)\) and that \(E\left(Y_{2}\right)=\mu_{2}\) and \(\operatorname{Var}\left(Y_{2}\right)=\sigma_{2}^{2}\). Assume that 𝑌 is a random variable whose density is a mixture of the densities corresponding to \(Y_{1} \text { and } Y_{2}\). Show that

i \(E(Y)=a \mu_{1}+(1-\mathrm{a}) \mu_{2}\).

ii \(\operatorname{Var}(Y)=a \sigma_{1}^{2}+(1-a) \sigma_{2}^{2}+a(1-a)\left[\mu_{1}-\mu_{2}\right]^{2}\).

[Hint: \(E\left(Y_i^2\right)=\mu_i^2+\sigma_i^2,\ i=1,2\).]

Equation Transcription:

Text Transcription:

f_1(y) and f_2(y)

0</=a</=1

f(y)=af_1(y)+(1-a)f_2(y)

f(y)

Y_1

f_1(y)

E(Y_1)=mu_1

Var(Y_1)=sigma_1^2

Y_2

f_2(y)

E(Y_2)=mu_2

Var(Y_2)=sigma_2^2

Y_1 and Y_2

E(Y)=amu_1+(1-a)mu_2

Var(Y)=asigma_1^2+(1-a)sigma_2^2+a(1-a)[mu_1-mu_2]^2

E(Y_i^2)=mu_i^2+sigma_i^2,i=1,2