Chapter 4. Probability Distributions Continuous

Transcription

1 1 Chapter 4 Probability Distributions Continuous Thus far, we have considered discrete pdfs (sometimes called probability mass functions) and have seen how that probability of X equaling a single number is clustered at a particular point. However, we can think of probability densities defined over an interval or a continuum of points. Here is an example. Suppose that we consider the following density f( s) = (4 s) / 8 for 0 s 4 and zero elsewhere The graph of this function shown in Figure 1 below. Of course, it is merely a triangle. The area of this triangle is the Area = (base)(height) /. But, the base is just 4 and the height is 1/. So, the area is exactly 1. This is needed, if f(s) is to be a density. The other condition is that f(s) must be greater than or equal to 0. But, that also holds, so we know that f(s) is a probability density. Note how that the support in this case is the interval [0,4]. That is, f(s) is positive for every number in [0,4]. This means that the density written down above is a continuous density and not a discrete density. Figure 1

2 Now considering that this is a continuous probability density, we can use it to compute probabilities for the random variable X that it describes. For example, suppose that we wanted to find the probability that X is less than some number α. This probability we write as PX [ α] and we see that it is just the area under the density from 0 to α. It is really just that simple. Figure shows this area in yellow. Figure To actually find the area in yellow in Figure is a little complicated. Usually we write it as an integral, which is a fancy way of describing the yellow area. Formally we would find this area by the process of integration. Here is how it is done. You can actually find the yellow area without doing the integration because the density is linear. Typically, it will be difficult to find areas like the yellow area in Figure. In some cases, an exact formula for the areas of certain densities is impossible and we must rely on printed tables. We will have a chance to look at these later. For now the integration is done as follows α α α α α P[ X α] = f ( s) ds = (4 s) ds = ( s) ds 8 = Therefore,

3 3 α α PX [ α] = 16 For all 0 α 4 and is equal to 0 for α < 0 and is equal to 1 for α > 1. An example of an exact calculation would be α = 3. Then, for this density, the probability that X is less than or equal to 3 is PX [ 3] = (if you have an Android phone you can download MathAlly from Google Play and actually do the integration on your phone). If you have a computer, you can do the integration here and get COMPARE Which is the same as our answer above forα = x and constant = 0. If you have an iphone you may want to invest in the following on the APP Store Less than $4 USD and it does integration well. The same company produces an Android version (very sophisticated) for only $98. It is called Wolfram Alpha and you can get it at Google Play. I bought it and worked the above problem on it and here is the screen shot from my phone

4 4 Here is the answer to our problem I think the Mathally is sufficient for your purposes and it is FREE. But, Wolfram is so cheap and it has so many great features that it is hard to turn down. Okay, so now we have computer tools that will let us do some very complicated integrations. We don t even need a calculus course to do these. We only need to know how to use the computer (or phone) to get our answer. Let's do some practice at finding probabilities by integration of densities. But, first let's look at the Normal density.

5 5 The Normal density is the most famous probability density in the world. Here is the formula 1 ( s µ ) σ f( s) = 1 e for - < s < πσ You should not be scared of this formula. Your phone can do any integration you need, once you put in the right information. The term π should not scare you. It is roughly equal to /7 (actually a little smaller). The term e is a number between and 3. It is roughly e = The two parameters ( µ and σ ) are unknown constants that we can input (later we will use data and estimate these). The first parameter µ is called the mean, while σ is called the variance (we call σ the standard deviation and note that σ = σ ). These parameters are fixed numbers. They do not change. They have nothing to do with data, except that we can use data on X to guess or estimate these two constants. If X is a random variable with this pdf then once we know µ and σ we can make all sorts of calculations about X's probabilities. Everything works just like the first example in Figure 1 and Figure. Figure 3 below shows a Normal density's graph. It has two long tails and is defined over the entire real line of numbers. This graph is drawn assuming that µ = 0 and Figure 3 σ = 1. It is the very special Standard Normal density which is usually written N(0,1). Please understand that not all normally distributed random variables have the standard normal

6 6 density. The only requirement is that σ is positive. The mean µ can be any number, whatsoever. Some people call the Normal density the bell-shaped curve since it looks like a bell. The area under the bell is equal to 1, since it is a density. About 68% of the area lies between -1 and 1. About 95% of the area lies between and -. Suppose we want to compute the probability that X 1.5. This is denoted PX [ 1.5]. Here is how we compute it (again we can use our phone and the software to get the answer using the substitution - 90 instead of ). Figure 4 For any continuous probability density function (pdf), say f() s, the cumulative distribution function (cdf), F(α ), is defined as α F( α) = f () s ds

7 7 (P1) Take the continuous pdf using your calculator or computer and integration software. (P) Take the continuous pdf 1 1 f( s) = ( s) for 0 s 4. Find the PX [ 0.5] 8 f s = e for 0 < s <. Find the probability 5s () 5 PX [ 0.] using your calculator or computer and integration software. (Note that this pdf is called the Negative Exponential pdf) (P3) Use your software on your phone or computer to show that for the continuous pdf 1 s (0 + ) 1 f( s) = A(1 + ) for - < s <, A. The constant A is called a normalizing constant since it makes the integral equal to 1 over the entire domain. You can substitute -60 for negative infinity and 60 for infinity in the formula when using your phone software. This type of pdf is called Student's t distribution with 0 degrees of freedom. By the way, a t distribution with 0 degrees of freedom will have a different normalizing constant A than one with 19 degrees of freedom, or one with 1 degrees of freedom, and so on. Not hard to find though, just replace 0 by 19 or 1 in the formula above and then find the constant that causes the integral to equal 1. (P4) Using the pdf in (P3) find PX [ 1.5]. (P5) Consider a Chi-Square χ density with 4 degrees of freedom. This can be written as 4 x ( 1) f ( s) = Ax e for 0 < x < Use your phone software to calculate A. You can replace infinity with 90 and then find that A = 0.5. Now, using this χ density calculate PX [ 1.5].