# Discrete uniform distribution prrof

• synkk
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synkk

Hello, I'm currently in high school and going over discrete uniform distribution, and we've come across this formula. I'm curious if anyone could show me how the formula is true, as when I asked my teacher he just said that it'll confuse the class and we don't need to know why it's true.

If anyone could show me a proof or something i'd be very grateful :)

synkk said:

Hello, I'm currently in high school and going over discrete uniform distribution, and we've come across this formula. I'm curious if anyone could show me how the formula is true, as when I asked my teacher he just said that it'll confuse the class and we don't need to know why it's true.

If anyone could show me a proof or something i'd be very grateful :)

You can do it directly if you know formulas for $\sum_{k=1}^n k \text{ and } \sum_{k=1}^n k^2,$ and these can be found on-line, for example. Another way is to prove the results by induction (although I don't know if you have studied that, yet).

Let's just do it directly for the E(X). The probability mass function is $p(k) = \Pr \{X=k\} = 1/n,$ for k = 1, 2, ..., n . The expected value is *defined* as $$E(X) = 1\cdot p(1) + 2 \cdot p(2) + 3 \cdot p(3) + \cdots + n \cdot p(n) = \frac{1}{n}[1 + 2 + \cdots + n].$$ This last summation is $$1+2+ \cdots +n = \frac{n(n+1)}{2},$$ so we we get the stated result.

Getting $\text{Var}(X)$ is more complicated, but you can use the easily-proven fact that $\text{Var}(X) = E(X^2) - (EX)^2,$ and so reduce the problem to finding $$E(X^2) = p(1) \cdot 1^1 + p(2) \cdot 2^2 + \cdots + p(n) \cdot n^2 = \frac{1}{n} [ 1^2 + 2^2 + \cdots n^2].$$

RGV

## 1. What is a discrete uniform distribution?

A discrete uniform distribution is a probability distribution in which a finite number of outcomes have an equal probability of occurring. It is often used to model situations where each outcome is equally likely, such as rolling a fair die or flipping a coin.

## 2. What is the formula for calculating the probability of a specific outcome in a discrete uniform distribution?

The probability of a specific outcome in a discrete uniform distribution is calculated by dividing 1 by the total number of outcomes. This can be expressed as P(X=x) = 1/n, where n is the number of possible outcomes.

## 3. How is the mean calculated in a discrete uniform distribution?

The mean in a discrete uniform distribution is calculated by taking the average of all possible outcomes. In other words, it is the sum of all outcomes divided by the total number of outcomes. This can be expressed as E(X) = (n+1)/2, where n is the number of possible outcomes.

## 4. What is the difference between a discrete uniform distribution and a continuous uniform distribution?

The main difference between a discrete uniform distribution and a continuous uniform distribution is that the former is used for a finite number of outcomes with equal probabilities, while the latter is used for a continuous range of outcomes with equal probabilities. For example, rolling a die would follow a discrete uniform distribution, while measuring the height of individuals in a population would follow a continuous uniform distribution.

## 5. How is the variance calculated in a discrete uniform distribution?

The variance in a discrete uniform distribution is calculated by taking the sum of the squared differences between each outcome and the mean, divided by the total number of outcomes. This can be expressed as Var(X) = (n^2 - 1)/12, where n is the number of possible outcomes.

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