Kurtosis of uniform distribution

Question

I am a beginner in statistics, and am self-studying. I want to determine the kurtosis for uniform distribution. Could someone please help me with this problem?

Answer 1

Just plug in the definition to find the kurtosis is $9/5$, where the mean is $\mu = 1/2$:

$$\kappa = {\int\limits_{x=0}^1 (x - \mu)^4 dx \over \left( \int\limits_{x=0}^1 (x - \mu)^2 dx \right)^2} = {\int\limits_{x=0}^1 (x-1/2)^4 dx \over \left(\int\limits_{x=0}^1 (x-1/2)^2 dx \right)^2} = {9 \over 5}$$

Answer 2

You can actually determine the kurtosis with a formula. The kurtosis of a Probability Density Function of a random variable $X$ is defined as:

$\beta_2=\frac{\mu_4(X)}{\mu_2(X)^2}=\frac{\mu_4(X)}{\sigma^4(X)} = \frac{\operatorname{E}[(X-\mu)^4]}{(\operatorname{E}[(X-\mu)^2])^2}$

with the expected value $\mu =\operatorname{E}[X]$

And

$\mu_k$

is by the way defined as:

$\mu_k := \operatorname{E}\left(\left(X-\mu\right)^k\right)$

Whereas

$\mu_4$

the Kurtosis and

$\mu_2$

the Variance is.

Thus consider:

$\beta_2 = \tfrac{9}{5} = 1{,}8$