Next: 1.4 – The Cumulative Distribution Function
Usually we are interested in experiments where there is more than one outcome, each having a possibly different probability. The probability density function of a discrete random variable is simply the collection of all these probabilities.
Discrete Probability Density Function |
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The discrete probability density function (PDF) of a discrete random variable X can be represented in a table, graph, or formula, and provides the probabilities Pr(X = x) for all possible values of x. |
Example: Different Coloured Balls
Although it is usually more convenient to work with random variables that assume numerical values, this need not always be the case. Suppose that a box contains 10 balls:
- 5 of the balls are red
- 2 of the balls are green
- 2 of the balls are blue
- 1 ball is yellow
Suppose we take one ball out of the box. Let X be the random variable that represents the colour of the ball. As 5 of the balls are red, and there are 10 balls, the probability that a red ball is drawn from the box is Pr(X = Red) = 5/10 = 1/2.
Similarly, there are 2 green balls, so the probability that X is green is 2/10. Similar calculations for the other colours yields the probability density function given by the following table.
Ball Colour | Probability |
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red | 5/10 |
green | 2/10 |
blue | 2/10 |
yellow | 1/10 |
Example: A Six-Sided Die
Consider again the experiment of rolling a six-sided die. A six-sided die can land on any of its six faces, so that a single experiment has six possible outcomes.
For a "fair die", we anticipate getting each of the results with an equal probability, i.e. if we were to repeat the same experiment many times, we would expect that, on average, the six possible events would occur with similar frequencies (we say that such events are uniformly distributed).
There are six possible outcomes: 1, 2, 3, 4, 5, or 6. The probability density function could be given by the following table.
Outcome | Probability |
---|---|
1 | 1/6 |
2 | 1/6 |
3 | 1/6 |
4 | 1/6 |
5 | 1/6 |
6 | 1/6 |
The PDF could also be given by the equation Pr(X = k) = 1/6, for k = 1, 2, 3, ... , 6, where X denotes the random variable associated to rolling a fair die once. Thus we see that uniform random variables have PDFs which are particularly easy to represent.
Next: 1.4 – The Cumulative Distribution Function