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September 11th, 2016, 09:56 AM   #1
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number of all positive functions

Nymber of all positive continuous function $f(x)$ in $\left[0,1\right]$ which satisfy $\displaystyle \int^{1}_{0}f(x)dx=1$ and $\displaystyle \int^{1}_{0}xf(x)dx=\alpha$ and $\displaystyle \int^{1}_{0}x^2f(x)dx=\alpha^2$

Where $\alpha$ is a given real numbers.

What I have tried :: adding (1) and (3) and subtracting (2), we. Get $\displaystyle \int^{1}_{0}(x-1)^2f(x)dx=(\alpha-1)^2$ now how can I solve it after that, Thanks
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September 11th, 2016, 10:59 AM   #2
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we're looking for a probability density function with

a) support on [0,1]

b) mean $\alpha$

c) variance = $E[x^2]-(E[x])^2 = \alpha^2 - \alpha^2 = 0$

The only function I know of that will have zero variance and mean $\alpha$ is $\delta(x-\alpha),~\alpha \in [0,1]$

$\delta(x-\alpha)$ also satisfies the first condition provide $\alpha$ is as specified.

So it looks like there is 1 function that satisfies all your criteria.
Thanks from panky
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