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

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.
  • #1
laura_a
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0

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?


Homework Equations





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...

Thanks for any help you can give me..
 
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  • #2
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.
 

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