Question

Let X has the discrete uniform distribution
f
(
x
)
=
1
k
f(x)=1k
for x=0,1,2…. , find the expression for its mean u

Answer 1

Answer on Question #68039 - Math - Statistics and Probability

Question

Let $X$ has the discrete uniform distribution $f(x) = \frac{1}{k}$ for $x = 0,1,2\ldots$, find the expression for its mean $u$.

Solution

In *probability theory* and *statistics* by discrete uniform distribution we denote a *probability distribution* whereby a finite number of values are equally likely to be observed; every one of $k$ values has equal probability $1 / k$.

Therefore, the given distribution is the discrete uniform distribution, if we write

$$f(x) = \frac{1}{k} \quad \text{for } x = 0,1,2,\ldots,k-1.$$

Then its mean is equal to

$$u = 0 \cdot \frac{1}{k} + 1 \cdot \frac{1}{k} + 2 \cdot \frac{1}{k} + \cdots + (k-1) \cdot \frac{1}{k} = \frac{1}{k} \cdot (0 + 1 + 2 + \cdots + k - 1).$$

In brackets we have a sum of arithmetic sequence. Therefore,

$$u = \frac{1}{k} \cdot \frac{(0 + k - 1)}{2} \cdot k = \frac{k - 1}{2}.$$

Answer: $u = \frac{k - 1}{2}$.

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