# Probability distribution explanation

What exactly is a probability distribution, and what are the two requirements for a probability distribution?

I am not sure what this means or how to apply it? Any examples that can be given would be great!

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Probability distributions describe the behavior of random variables. Informally a random variable is a variable whose values occur by some chance mechanism. Probability distributions describe that mechanism. The cumulative probability function $F(x)=P[X\le x]$ when $x$ is a real number and $X$ is a real valued random variable where $P[X\le x]$ reads as the probability that $X$ is less than or equal to $x$. $0\le F(x)\le 1$ and for any $y<x$, $F(y)\le F(x)$. If $F$ is absolutely continuous, it has a derivative $f(x)=F'(x)$ which is called the probability density. For every $x$, $f(x)\ge0$ and over the range of $X$ it integrates to $1$. If the distribution is discrete meaning that it takes on an at most countable set of values $x_i$ then $F(x)= \sum p(x_i)$ where $p(x_i) =$ probability that $X=x_i$, and the sum is taken for all $x_i\le x$. $F$ can be a mixture of discrete and absolutely continuous components among other possibilities for a non decreasing non negative function that increases to $1$.

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The cumulative distribution function is a nondecreasing function $F(x)$ that is continuous from the right, with $F(x) \to 0$ as $x \to -\infty$ and $F(x) \to 1$ as $x \to +\infty$. – Robert Israel Jun 22 '12 at 21:47
I see in Wikipedia that the cumulative distribution function is given as $\mathrm{P}[X\le x]$, but is that standard, or could it alternately be defined as $\mathrm{P}[X\lt x]$? – robjohn Jun 22 '12 at 22:13
I think <= is standard. I see no harm in defining it the other way though. – Michael Chernick Jun 22 '12 at 22:23