# Uniform Distribution over n and its limit

• JonoPUH
In summary, we are looking at the distribution of Yn/n, which is the random variable Yn divided by n. The distribution function of Yn/n is given by Gn(x) and can be calculated by plugging in nx for x in the distribution function of Yn. As n approaches infinity, the sequence Yn/n converges in distribution to a limit of 0.
JonoPUH

## Homework Statement

Let Yn be uniform on {1, 2, . . . , n} (i.e. taking each value with probability 1/n). Draw the distribution function of Yn/n. Show that the sequence Yn/n converges in distribution as n → ∞. What is the limit?

## Homework Equations

So Yn has c.d.f Yn(x) = |x|/n where |x| is the nearest integer less than x.

## The Attempt at a Solution

Is it ok for me to just divide the c.d.f of Yn by n, and so plot |x|/n2?
This seems far too easy for 2nd year of university (I really do not trust myself with probability) and also it only gives a total for x > n of 1/n. Is this allowed?

If so, then Yn/n clearly tends to 0 as n → 0 since we have 1/n.

Thanks

Last edited:
JonoPUH said:

## Homework Statement

Let Yn be uniform on {1, 2, . . . , n} (i.e. taking each value with probability 1/n). Draw the distribution function of Yn/n. Show that the sequence Yn/n converges in distribution as n → ∞. What is the limit?

## Homework Equations

So Yn has c.d.f Yn(x) = |x|/n where |x| is the nearest integer less than x.

## The Attempt at a Solution

Is it ok for me to just divide the c.d.f of Yn by n, and so plot |x|/n2?
This seems far too easy for 2nd year of university (I really do not trust myself with probability) and also it only gives a total for x > n of 1/n. Is this allowed?

If so, then Yn/n clearly tends to 0 as n → 0 since we have 1/n.

Thanks

Your notation is horrible and will get you into trouble. Yn is a random variable, not a function of x. Its distribution function is $$F_n(x) = P(Y_n \leq x) = \left\{ \begin{array}{cl}\frac{\lfloor x \rfloor}{n},& \: 0 \leq x < n\\ 1,&\: x \geq n \end{array}\right.$$ as you said. The distribution of Y_n/n is
$$G_n(x) = P\left( \frac{Y_n}{n} \leq x \right) = P( Y_n \leq n x ) = F_n(nx) = \left\{ \begin{array}{cl} \frac{\lfloor nx \rfloor}{n}, &\: 0 \leq nx < n\\ 1, & \: nx \geq n \end{array} \right.$$

Thank you for your clarification on notation and help!

## 1. What is a uniform distribution over n and its limit?

A uniform distribution over n is a probability distribution where all possible outcomes have an equal chance of occurring. Its limit is the maximum value that the distribution can take, which is equal to 1.

## 2. How is a uniform distribution over n different from other types of distributions?

A uniform distribution over n is different from other distributions because it has a constant probability of occurring for all possible outcomes, while other distributions have varying probabilities for different outcomes.

## 3. How is a uniform distribution over n used in scientific research?

A uniform distribution over n is commonly used in scientific research to model situations where all outcomes are equally likely, such as in random sampling or in the study of genetics.

## 4. Can a uniform distribution over n have a negative value?

No, a uniform distribution over n cannot have a negative value since it represents a probability and probabilities cannot be negative.

## 5. How is the limit of a uniform distribution over n calculated?

The limit of a uniform distribution over n is calculated by dividing 1 (the maximum value of the distribution) by the number of outcomes (n).

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