# Finding Distribution Function for p(x) = log_10(1 + 1/x)

• laura_a
In summary, the question is about proving that p(x) is a probability function and determining an expression for the corresponding distribution function, which is the sum of all p(y) where 1 <= y <= x. The asker has difficulties understanding the concept and formatting of the distribution function, and is seeking guidance on how to write a general expression.
laura_a

## Homework Statement

I have the following question

p(x) = log_10(1 + 1/x) for x = 1,2,3, ... 9 (otherwise p(x)=0)

So firstly I had to prove that p(x) is a probability function, which I have done so (by proving the sum of all the values =1)
anyway the second thing I have to do is determine an expression for the corresponding distribution function. How exactly do I do this? The information on wiki is confusing and the textbook doesn't seem to cover it? Can anyone help?

## The Attempt at a Solution

I've read 2 textbooks and the internet and I can't find an exact method or style of answer, but I know it has to have inequalities in it and I have done some working out plugging in each value and getting an answer, but I am unable to write it out in the format that the textbook has... that is would be something like

P_x(x) = { 0 for x <1
... etc... but if I wrote something like =1 for x<=9 that isn't exactly true because if x was 8.5 then it wouldn't equal one... this is what I was just working on when I realized I must have the wrong idea... the question asks me to write an expression for the distribution function... is there a simple way I can just write a general?? Do I need to interal because I have worked that out just in case...

What you're working with here is a discrete probability function, as you probably know. For each x you get the probability for that outcome. The random variable presumably denoted X has for example a chance log10(1 + 1/1) = log10(2) of being 1, and so on for the other defined values of x.

A distribution function, on the other hand, doesn't give you the probability that X is equal to x, but the probability that the outcome of X is less than or equal, to the given x. In this case, it's just the sum of all p(y) where 1 <= y <= x.

## What is the distribution function for p(x) = log10(1 + 1/x)?

The distribution function for p(x) = log10(1 + 1/x) is F(x) = log10(1 + 1/x) - log10x.

## What is the domain of the distribution function?

The domain of the distribution function is all positive real numbers, as x must be greater than 0 for the function to be defined.

## What is the range of the distribution function?

The range of the distribution function is all real numbers between 0 and 1, inclusive.

## What is the behavior of the distribution function as x approaches 0?

As x approaches 0, the distribution function approaches 0. This is because as x gets closer to 0, the log10(1 + 1/x) term becomes smaller and smaller, while the log10x term stays constant at 0. Therefore, the overall value of the function approaches 0.

## What is the behavior of the distribution function as x approaches infinity?

As x approaches infinity, the distribution function approaches 1. This is because as x gets larger and larger, the log10(1 + 1/x) term becomes smaller and smaller, while the log10x term also gets larger. Therefore, the overall value of the function approaches 1.

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