Kurtosis poisson

Question

I am trying to find the kurtosis of the Poisson - I found

E(X ⁴)) = λ⁴ + 6λ³ + 7λ² + λ`

The 4^th central moment I calculated with the binomial expansion:

E(X ⁴)) - 4 E(X ³)) E(X) + 6E(X ²)) E(X)² - 4E(X)E(X)³ + E(X)⁴

= λ⁴ + 6λ³ + 7λ² + λ - 4λ(λ³ + 3λ² + λ) + 6λ²(λ² + λ) - 4λ⁴ + λ⁴

However, I got 4λ²/λ² when dividing it with sigma⁴ - according to a solution I got I should be 1/λ.

Answer 1

Your calculation of $\mathsf E[X^3]$ is not correct. Another answer derives it as

$$\mathsf E[X^3]=\lambda^3 + \color{red}3\lambda^2 + \lambda$$