Math Help Forum - View Single Post - (Cumulative) Probability Density Functions

artvandalay11 · #2 November 5th, 2009, 08:51 PM

Quote:

Originally Posted by ferriswheel

Hey everyone, ill make this short and sweet if i can.

I am trying to understand what a probability density function is, and then understand what a cumulative probability density function is; and also what the area under a probability density function is.

My professor has been...less then helpful and the internet hasn't been too much of a help either. I really appreciate your help. Thanks.

A cumulative probability density function is meaningless.... do you mean(CDF) comulative distribution function?

If so, this is a function which takes inputs and spits out outputs (as all functions do). The outputs though, are the probabilities that the Random Variable is $\leq$ the value you input

Example, for the Normal CDF with mean 0 and standard deviation 1, if you plug in 1.32, you are going to get the Probability that Z is less than or equal to 1.32 given that Z is normal with mean 0 and standard deviation 1

A probability density function (PDF) is another function, with different properties. Firstly, if you integrate it, you get the CDF. If you integrate it over all values, you must get 1 since the sum of probabilities must =1. You use this function to find what the Probability is of a random variable being between 2 numbers. (integrate between the two numbers)

I'm not sure you're looking for the rigorous definition. If so, you can simply take it to be the derivative of the CDF to avoid headaches.

The difference between the two should be obvious, as the PDF is the derivative of the CDF, but the properties are key. The CDF evaluated at $-\infty$ is 0, since the probability of being less than $-\infty$ is 0.

Similarly, the CDF evaluated at $\infty$ is 1. (when I say evaluated at infinity, you of course use limits, you cannot actually plug in infinity)

The CDF is also non-decreasing, which is obvious if you think about the definition in terms of probability. If you take a bigger value to plug in, the probability that the random variable is smaller cannot possibly be less than it was at the value before (If $y_1<y_2$ then $F(y_1)\leq F(y_2)$ )

The PDF is always greater than or equal to zero since the CDF is non-decreasing and the PDF is the the derivative of the CDF. Also the integral from $-\infty \text{ to } \infty=1$ also because the PDF is the derivative of the CDF

I hope this helps somewhat. Many sites explain it with more math and much more rigorously, but I used english since your confused. There is no cumulative denisty function though, there is a cumulative distribution function and a density function