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Moment of Density Problem... midterm in 2 hours helpppp
If f is a nonnegative function whose integral is equal to 1, then f defines a probability density; the kth moment of this distribution is defined to be the average value of x^k with respect to this density. Compute all moments of the density defined by f(x) = e^(-x) on the positive half-line.
\begin{displaymath}M(\theta) = E[e^{X\theta}] = \int_{-\infty}^{\infty} e^{x\theta} f(x) dx. \end{displaymath}
The kth central moment of a random variable X is given by E[(X-E[X])k
The answer is K!, but i don't know how to get there.
THANKS SO MUCH!
Homework Statement
If f is a nonnegative function whose integral is equal to 1, then f defines a probability density; the kth moment of this distribution is defined to be the average value of x^k with respect to this density. Compute all moments of the density defined by f(x) = e^(-x) on the positive half-line.
Homework Equations
\begin{displaymath}M(\theta) = E[e^{X\theta}] = \int_{-\infty}^{\infty} e^{x\theta} f(x) dx. \end{displaymath}
The kth central moment of a random variable X is given by E[(X-E[X])k
The Attempt at a Solution
The answer is K!, but i don't know how to get there.
THANKS SO MUCH!