Distribution Functions A distribution function is a function which describes how values are allocated across a population or sample space. There are a variety of settings for which such functions arise and we shall turn our attention to some examples. Example 1: Suppose you flip a fair coin ten times. What is the distribution of the number of heads that you will observe? Plot the distribution. Solution: There are a total of 1024 different combinations of heads and tails that we can observe since we have ten slots to fill and in each slot, there are only two possible values, a Head or a Tail. So, we have 2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2ÿ2 = 210 = 1024 different outcomes. We could count each of the 1,024 outcomes and record the number of heads. This would take a little while and in the interest of space, the details are omitted. Instead, we end up with the following table: Number of Heads 0 2 3 4 5 6 7 8 9 10Proportion showing56 330 462 165 11that many heads1024 1024 1024 1024 1024 We can also display our data in a histogram. A histogram is a graphical way to represent data where vertical bars are paced above each category (group) in such a way that the area of each bar represents the proportion of the population in that category. Figure 1: Histogram of Coin Flips 1Notice that we grouped the data in only five categories. If we had instead made a table for each coin flip, we could get better estimates. And we might be interested in fitting a smooth curve to the data. The curve that we fit should have the property that the area under the curve above one group is equal to the area of the corresponding rectangle. Doing this, we have the following figure. Figure 2: Histogram of Coin Flips with a Smooth Curve The smooth function that we applied to the histogram above is called a (probability) density function. This function has the property that Proportion of population for Area under the graph of ()which is between and ( ) between and bapxdxxabpxab If a and b are the smallest and largest values that our data can take. (In this case, the smallest number of heads we can observe is 0 and the largest is 10), then we have that 100() () 1bapxdx pxdx So far, we have only considered the integral of p(x) and not the function p(x) itself. Looking at Figure 2, notice that when x = 5, curve is approximately equal to 0.25. This does not mean that 0.25 of the coin flips will come up with 5 heads, in general. Rather, we interpret p(5) = 0.25 to mean that for some all interval, Dx, around 5, the proportion of the samples with number of heads in this interval is approximately equal to p(5)Dx = 0.25Dx. 2(Probability) density functions also have to have two other important properties. First, they must be non-negative. This is because the integral always gives a proportion of the population, which itself must be a positive number between 0 and 1. Second, the integral of p(x) from -¶ to ¶ is equal to 1. This is because in the interval (-¶, ¶), we will have observed everything possible. Therefore, the proportion of population will be equal to 1. We record these properties of a (probability) density function in the box below. (Probability) Density Function A function p(x) is called a (probability) density function if Proportion of population for Area under the graph of ()which is between and ( ) between and bapxdxxabpxab and () 1pxdx and 0 § p(x) § 1 for all x Example 2: Suppose a real number is chosen at random on the interval [1, 6]. Such a density function is known as a uniform distribution, since all of the points are equally likely to be chosen. The graph of this function is as follows: 1 2 3 4 5 6xcpx Figure 3: A uniform distribution on [1, 6] Find the value of c which will make this a density function. Solution: Since the area under the curve needs to equal 1 and we are dealing with a rectangular area with a base of 6 – 1 = 5 and a height of c, we have the equation 5c = 1. Solving for c, we see that c = 1/5. 3Another way of discussing how values are distributed is by using what is known as a cumulative distribution function. It is defined as the function Proportion of population() ( )having values of below tPt pxdxxt where p(x) is the (probability) density function discussed above. Notice that by the Fundamental Theorem of Calculus, the function P(t) is the antiderivative of p(x), with lim ( ) 0tPt. We use the word cumulative to denote the fact that this function measures the area under the curve from -¶ to t. Returning to Example 2, the shaded region in the next figure shows the proportion of numbers that are between 1 and 3. 1 2 3 4 5 6x0.2px Figure 4: Proportion of numbers between 1 and 3 The cumulative distribution function that corresponds to the shaded areas is given by the following graph: 1 2 3 4 5 6t0.51Pt Figure 5: Proportion of numbers less than t In particular, notice that the graph is a constant value of 1 after t = 6. This corresponds to the fact that after 6, we have seen all of the possible numbers that we could randomly select. We record these properties of a cumulative density function in the box below. 4Cumulative Distribution Function A function P(t) is called a cumulative distribution function if Proportion of population() ( )having values of below tPt pxdxxt And it satisfies the following properties 1. P(t) is a non-decreasing function 2. l and im ( ) 1tPt lim ( ) 0tPt 3. Proportion of population having() () ()values of between and bapxdx Pb Paxab Example 3: Determine if the following graphs correspond to a probability density function or a cumulative distribution function and find the value of c. 4xc cx2 Solution: The first graph corresponds to a cumulative distribution function since the graph reaches a plateau and remains there. Moreover, since lim ( ) 1xPx, we see that c = 1. The second graph corresponds to a probability density function. Recall, the area under the curve must be equal to 1. Here, we have a triangle with base c and height 2. The area of the triangle is thus (1/2)ÿ2ÿc, so we see that c = 1 as well.