- #1

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## Homework Statement

Finding the expected value of x, with poisson distribution. I don't follow the sum. It goes like this:

[tex] E(x)= \sum_{x=0}^{\infty} \frac{xe^{-\lambda}\lambda^x}{x!} [/tex]

[tex] = e^{-\lambda} \sum_{x=0}^{\infty} \frac{x\lambda^x}{x(x-1)!} [/tex]

[tex] = \lambda e^{-\lambda} \sum_{x=1}^{\infty} \frac{\lambda^{x-1}}{(x-1)!} [/tex]

[tex] = \lambda e^{-\lambda} \sum_{k=0}^{\infty} \frac{\lambda^{k}}{k!} [/tex]

[tex] = \lambda e^{-\lambda}e^{\lambda} = \lambda [/tex]

So basically the part I don't get is why they say

[tex] \sum_{k=0}^{\infty} \frac{\lambda^{k}}{k!} = e^{\lambda} [/tex]