Solution Found!
Let f1(y) and f2(y) be density functions and let a be a
Chapter 4, Problem 185SE(choose chapter or problem)
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
Questions & Answers
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
ANSWER:
Solution
Step 1 of 2
a) We have to show that f(y) is a density function when it is a mixture of two density functions
If f(x) is density function we have to show that
Given that and is a density functions
then
And
Given that
Now
=
=
=1
Hence prove that
So, that f(y) is a density function