Question 1203892
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To find the expected value, we apply the integral to x*f(x) over the limits from -infinity to infinity.


E(x) = expected value
*[tex \Large E(\text{x}) = \int_{-\infty}^{\infty}\text{x}*f(\text{x})d\text{x}]


Because much of the real number line makes f(x) = 0, we only need to worry about the interval {{{0 < x < 1/(sqrt(theta))}}} which is the same as {{{0 < x < sqrt(1/theta)}}} where theta > 0


The goal has been reduced to computing 
*[tex \Large E(\text{x}) = \int_{0}^{\sqrt{1/\theta}}\text{x}*f(\text{x})d\text{x}]


Using fairly elementary calculus integration rules, you should find that:
*[tex \Large g(\text{x}) = \int \text{x}*f(\text{x})d\text{x}]
*[tex \Large g(\text{x}) = \int \text{x}*(\theta*\text{x} + (3/2)\theta^{3/2}\text{x}^2)d\text{x}]
*[tex \Large g(\text{x}) = \int (\theta*\text{x}^2 + (3/2)\theta^{3/2}x^3)d\text{x}]
*[tex \Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C]
For some constant C.


Then,
*[tex \Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C]
*[tex \Large g(0) = (1/3)\theta*0^3 + (3/8)\theta^{3/2}*0^4+C]
*[tex \Large g(0) = C]
and
*[tex \Large g(\text{x}) = (1/3)\theta*\text{x}^3 + (3/8)\theta^{3/2}\text{x}^4+C]
*[tex \Large g(1/\sqrt{\theta}) = (1/3)\theta*(1/\sqrt{\theta})^3 + (3/8)\theta^{3/2}*(1/\sqrt{\theta})^4+C]
*[tex \Large g(1/\sqrt{\theta}) = (1/3)\theta*\theta^{-3/2} + (3/8)\theta^{3/2}*\theta^{-2}+C]
*[tex \Large g(1/\sqrt{\theta}) = (1/3)\theta^{-1/2} + (3/8)\theta^{-1/2}+C]
*[tex \Large g(1/\sqrt{\theta}) = (1/3+3/8)\theta^{-1/2}+C]
*[tex \Large g(1/\sqrt{\theta}) = (17/24)\theta^{-1/2}+C]
*[tex \Large g(1/\sqrt{\theta}) = C + \frac{17}{24\sqrt{\theta}}]


Subtract the two results to find 
*[tex \Large E(\text{x}) = \int_{0}^{\sqrt{1/\theta}}\text{x}*f(\text{x})d\text{x}]
*[tex \Large E(\text{x}) = g(1/\sqrt{\theta}) - g(0)]
*[tex \Large E(\text{x}) = C + \frac{17}{24\sqrt{\theta}} - C]
*[tex \Large E(\text{x}) = \frac{17}{24\sqrt{\theta}}]


Therefore, the expected value is *[tex \Large \frac{17}{24\sqrt{\theta}}] where theta > 0
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