0123 2 2 x n x f x x x 1 for 123456 6 f x x when we

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0,1,2,3 2 2 x n x f x x x        1 ( ) for 1,2,3,4,5,6 6 f x x When we write out our pmf in functional form, we want to be sure to include the support. It should be noted that writing out the pmf in tabular form carries the exact same information as it does in functional form. We could also graph each pmf as done below. This format also carries the same information. Certainly, if the support was a larger set, this graph could be very useful. We purposely use dots with a line projected down. The probabilities hare are point masses and are not spread over an interval on the x-axis. There is a mass at 0, 1, 2 and 3. As always, the total mass (probability) is equal to one. In all three formats, the total probability of 1 is distributed over the support. We therefore refer the information about the support and associated probabilities as the Distribution . The stick figures, like the graph given above, carry more information when we have more than just a few values in our support. Consider the visualization of four random variables given below. The graph can contain much more intuitive information than we would get by looking at a table of values. While we will use the table of values to determine answers to probability questions, the graph can give us a feeling of the distribution. In each of the graphs we can instantly get a feeling for the distribution. If we looked at random variables with many more possible x-values, we might see even more information that might not be seen when looking at a table of values (see below). x f(x) 0 1/8 1 3/8 2 3/8 3 1/8 x f(x) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 6 5 4 3 2 1 0.22 0.20 0.18 0.16 0.14 0.12 C1 pmf of Rolling a Die 3 2 1 0 0.375 0.250 0.125 C4 pmf of Tossing 3 Coins and X = Number of Heads
5 The pmf is one way to convey the distribution (the story of probability) of a random variable. A second method for writing out the distribution is the Cumulative Distribution Function (CDF). Definition 3.5: For a random variable X , we define the Cumulative Distribution Function (CDF) as ( ) ( ) F x P X x . Example 3.6: P X F ( 5) (5) . If we had a CDF chart for X, we could read the answer from the chart. The information contained in the pmf and CDF are identical. Neither tells us more about the story of probability than the other, so both are considered the distribution. The distribution considered in its CDF format can be very handy for discrete random variables and is a necessity for continuous random variables that we will discuss in the next chapter. For our two previous examples, the CDFs are given below. We get from the pmf to the CDF by addition and from the CDF to the pmf by subtraction. When we study continuous distributions, can you guess how we get from one to the other? ________________ When we are discussing discrete random variables, a graph of the CDF is not as useful as the pmf. We can however see some things from these sample graphs.

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