Imum Coeli
Junior Member
- Joined
- Dec 3, 2012
- Messages
- 86
Hi, I have this question which tells me the answer I cannot get to. I was wondering if someone could tell me where I went wrong?
Question:
Suppose that \(\displaystyle Y_i \) is a random variable with probability density function \(\displaystyle f_i(y) \) and that \(\displaystyle E[Y_i] = \mu_i \) and \(\displaystyle var(Y_i) = \sigma_i^2 \) for \(\displaystyle i=1,\ldots,n\). Assume that the pdf \(\displaystyle f(y) \) of \(\displaystyle Y\) is
\(\displaystyle f(y)= \sum_{i=1}^n a_i f_i(y) \) for \(\displaystyle a \in (0,1), \) such that \(\displaystyle \sum_{i=1}^n a_i =1 \)
Given that \(\displaystyle E[Y] = \sum_{i=1}^n a_i \mu_i \) and \(\displaystyle E[Y_i^2] = \mu_i^2 + \sigma_i^2 \) show that for \(\displaystyle n=2 \) we have \(\displaystyle var(Y) = a_1 \sigma_1^2 + a_2 \sigma_2^2 + a_1a_2(\mu_1-\mu_2)^2.\)
Answer:
\(\displaystyle
\begin{aligned}
var(Y) &= E[Y^2]- E[Y]^2 \\
&= \sum_{i=1}^2a_i E[Y_i^2] - \left( \sum_{i=1}^2a_i \mu_i \right)^2 \\
&= \sum_{i=1}^2a_i (\mu_i^2 + \sigma_i^2) - \left( \sum_{i=1}^2a_i \mu_i \right)^2 \\
\implies \sum_{i=1}^2a_i \mu_i^2 - \left( \sum_{i=1}^2a_i \mu_i \right)^2 &= a_1a_2(\mu_1-\mu_2)^2
\end{aligned}
\)
which I'm pretty sure is incorrect...
Question:
Suppose that \(\displaystyle Y_i \) is a random variable with probability density function \(\displaystyle f_i(y) \) and that \(\displaystyle E[Y_i] = \mu_i \) and \(\displaystyle var(Y_i) = \sigma_i^2 \) for \(\displaystyle i=1,\ldots,n\). Assume that the pdf \(\displaystyle f(y) \) of \(\displaystyle Y\) is
\(\displaystyle f(y)= \sum_{i=1}^n a_i f_i(y) \) for \(\displaystyle a \in (0,1), \) such that \(\displaystyle \sum_{i=1}^n a_i =1 \)
Given that \(\displaystyle E[Y] = \sum_{i=1}^n a_i \mu_i \) and \(\displaystyle E[Y_i^2] = \mu_i^2 + \sigma_i^2 \) show that for \(\displaystyle n=2 \) we have \(\displaystyle var(Y) = a_1 \sigma_1^2 + a_2 \sigma_2^2 + a_1a_2(\mu_1-\mu_2)^2.\)
Answer:
\(\displaystyle
\begin{aligned}
var(Y) &= E[Y^2]- E[Y]^2 \\
&= \sum_{i=1}^2a_i E[Y_i^2] - \left( \sum_{i=1}^2a_i \mu_i \right)^2 \\
&= \sum_{i=1}^2a_i (\mu_i^2 + \sigma_i^2) - \left( \sum_{i=1}^2a_i \mu_i \right)^2 \\
\implies \sum_{i=1}^2a_i \mu_i^2 - \left( \sum_{i=1}^2a_i \mu_i \right)^2 &= a_1a_2(\mu_1-\mu_2)^2
\end{aligned}
\)
which I'm pretty sure is incorrect...