September 11th, 2016, 09:56 AM  #1 
Senior Member Joined: Jul 2011 Posts: 405 Thanks: 16  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 $\displaystyle \int^{1}_{0}(x1)^2f(x)dx=(\alpha1)^2$ now how can I solve it after that, Thanks 
September 11th, 2016, 10:59 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,529 Thanks: 1389 
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. 

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