ce is a small number. If you know p(x), you can compute variance using the following integral equation: (3.4) One last thing about variance. You can usually get a feel if it will be big or small by looking at a graph of p(x). If this graph is kind of flat and spread out over a lot of x values, variance will be big; if this graph is peaky, or not spread out over a lot of x values, then variance is small. Figure 3.2 shows you what I mean. Figure 3.2 (a) Random variable X with large variance (b) Random variable X with small variance A Review of Some Important Math, Stats, and Systems Example 3.1 Given a random variable X and told that X has a probability distribution function p(x) shown in Figure E3.1, determine its mean and its variance. Figure E3.1 The p(x) for our example We'll use equation (3.3) to get the mean like this: (E3.1) (E3.2) (E3.3) (E3.4) (E3.5) We'll use equation (3.4) to get the variance (it's plug-and-chug!) (E3.6) Chapter Three (E3.7) (E3.8) (E3.9) (E3.10) Oz=1/3 (E3.11) 3.1.5 Multiple Random Variables What do you do if you have more than one random variable? Let's say you have two random variables, X which is tomorrow's temperature, and y which is the temperature on the day after tomorrow. How do you describe these two random variables? That's usually handled using a joint probability density function p (x,y). This is just a simple extension of the probability density function p(x). Mathematically, p (x,y) is defined by the integral equation Plassis, (3.5) In words, you can use p(x,y) to tell how likely it is that (x,y) falls within any range of possible values. But why stop at two random variables? Let's say you give me 26 random variables a,b,c, right to Z, representing the temperature for the next 26 days. If you want to describe these statistically, you can characterize them with the joint probability density function p (a,b,c,...,2). This is defined by the integral equation db...dz (3.6) And so, just like the p(x) or p(x,y) before it, you can use p (a,b,c,...,2) to tell you how likely it is 