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I am confused about calculating the expected value of a function that is split, the question looks as follows:

question

and my solution is:

sol

Which are incorrect.. can someone please help me to correct the answers. Or tell me what I'm doing wrong. Thank you

Community
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amine
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  • Yes, you write $\int xf(x)dx$ but then you calculate $\int xF(x)dx$. Calculate first $f(x)$ from $F(x)$. – Jimmy R. Sep 30 '16 at 10:01

1 Answers1

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You have the correct equation for calculating the expectation of a continuous random variable from its probability density function.   However, that was not what you had been given.

You were given a Cumulative Distribution Function that belongs to a discrete random variable; and thus has a probability mass function.

$$p_X(x) =\begin{cases} 5/10 &:& x=0\\ 1/10 &:& x=1\\ 2/10 &:& x=2\\ 1/10 &:& x=3 \\ 1/10 & :& x=3.5 \\ 0 &:&\textsf{elsewhere} \end{cases}$$

As such the expectation is not an integral, but a series: $$\mathsf E(X) ~=~ \sum_{x\in\{0,1,2,3,3.5\}} x~p_X(x)$$

And similarly for the variance.

Graham Kemp
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