Every random variable is associated with a probability distribution that describes the variable completely. A probability function is one way to view a probability distribution. It specifies the probability that the random variable takes on a specific value; P(X = x) is the probability that a random variable X takes on the value x.
A probability function has two key properties:
The following examples will utilize these two properties in order to examine whether they are probability functions.
p(x) = x/6 for X = 1, 2, 3, and p(x) = 0 otherwise
p(x) = (2x - 3)/16 for X = 1, 2, 3, 4 and p(x) = 0 otherwise
Substituting into p(x): p(1) = -1/16
It is impossible for any probability to be negative, so it's not necessary to continue. Property 1 is violated, so it can be said straightaway that p(x) is not a probability function.
Note that individual probabilities in a continuous case cannot occur, so P(X = 5), say, is 0 if X is continuous.
In a continuous case, only a range of values can be considered (that is, 0 < X < 10), whereas in a discrete case, individual values have positive probabilities associated with them.
For a discrete random variable, the shorthand notation is p(x) = P(X = x). For continuous random variables, the probability function is denoted f(x) and called probability density function (pdf), or just the density. This function is effectively the continuous analogue of the discrete probability function p(x).
|wms3: f(x) is a probability density function pdf and it is the area under the function that determines probability so this mus be equal to 1. The line can go above one! p(x) is for discrete random variables and f(x) is for continuous.|
|stitcher: The probability density function measures an area under the curve. An area cannot be negative space|
|bidisha: Loving the stop here|
|tomalot: STOP HERE!|
| BossMan: p(x) = x/6 for x = 1, 2, 3 and p(x) = 0 otherwise.|
Does that mean that the outcome can ONLY be a 1, a 2 or a 3? Nothing else is possible?
|Olesya_CFA: @Bossman, right. The problem specifies that everything beyond 1, 2, and 3 gives us zero.|
|MathLoser: Remember to read that 4 dots at the end carefully, guys.|