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E[X4] = ∫∞ − ∞x4f(x)dx = 1 2πσ2∫∞ − ∞x4e − x2 2σ2dx. You can solve this integral by using partial integration a number of times. An alternative approach is to determine the moment generating function and differentiate. The moment generating function of a continuous random variable X is defined as. MX(t): = E[etX] = ∫∞ ...
Once you know what's going on, there is actually an easy algebraic trick to prove it: Var(X) = E(X2) − E(X)2 = 0 V a r (X) = E (X 2) − E (X) 2 = 0. – user14972. Dec 11, 2016 at 11:46. I see. Thanks a lot for clarifying this question! – Gyakenji. Dec 11, 2016 at 11:47. Add a comment.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Basically E[XY] = E[E[XY|Y]] = E[YE[X|Y]] E [X Y] = E [E [X Y | Y]] = E [Y E [X | Y]]. The first step is the iterated rule of conditional expectation. For the second, use the fact that given Y, Y is like a constant. However if you are looking for the usage of rigorous definition of conditional expectation, the solution by Davide Giraudo is the ...
Taking the limit as x → 0 x → 0 the 1/xk 1 / x k factor will head toward ∞ ∞, but the e1/x2 e 1 / x 2 will head for zero faster. In every case the exponential term will govern what happens and that term is going to zero, both from the right and left. So you can conclude that all derivatives exist and are 0 at x = 0.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Given X, Y: (Ω, F) → (R, B) two random variables on some probability space (Ω, F, P) where here F is the σ -algebra of events of Ω and B is the Borel σ -algebra of R. Then E[X + Y]: = ∫Ω(X(ω) + Y(ω))P(dω) = ∫ΩX(ω)P(dω) + ∫ΩY(ω)P(dω) = E[X] + E[Y] that is, the integral ∫ΩX(ω)P(dω) is linear. The x and y preceding the ...