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I need help in finding the variance of a joint probability function. The probability density function in this case is $f(x)$ which is created by $$\frac{125x^2}{18},\; 0\le x\le 0.6,\;\frac{9}{10x^2},\; (0.6\le x\le 0.9), \text{ and } 0 \text{ elsewhere}.$$ I know how to find the variance for a single probability function, however not when they consist of two functions.

Help will be appreciated. Thanks.

amWhy
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Deep Patel
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1 Answers1

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You are given a piecewise probability density function: $$f_X(x)=\begin{cases} 125 x^2/18 &:& ~~~0\leq x\lt 0.6\\ 9/(10x^2)&:& 0.6\leq x\leq 0.9\\0&:&\textrm{elsewhere}\end{cases}$$

To find an expected value, you simply integrate over the partitions using the relevant function and add: $$\mathsf E(g(X)) = \int_0^{0.6} 125x^2~g(x)/18~\mathsf dx + \int_{0.6}^{0.9} 9~g(x)/(10x^2)~\mathsf d x$$

Now evaluate: $\mathsf{Var}(X)=\mathsf E(X^2)-\mathsf E(X)^2$

Graham Kemp
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  • what is the g(x) in the E(X) for. What does it represent, i have never come across this before. Also, i have never come across the equation Var(X) = E(X^2) - E(X)^2 – Deep Patel Aug 29 '18 at 12:24
  • @DeepPatel $g(X)$ is a function of the random variable: substitute the function you require. $\mathsf {Var}(X)=\mathsf E(X)-\mathsf E(X)^2$ is the second equation for variance which you should memorise. Perhaps the first, depending on your teacher. The other is $\mathsf {Var}(X)=\mathsf E((X-\mathsf E(X))^2)$ – Graham Kemp Aug 29 '18 at 21:01