More About Expected Value and Variance Page 1 of 5 Expected value E X has a number of interesting properties These aren t likely to be used in this course beyond this lesson but may come into play in a later statistics course Properties of E X 1 E X is a linear operator By this we mean that things that add inside the add outside 2 E k k You could apply a Duh to this one The expected value for any constant outcome is the constant It comes directly from the definition whether discrete or continuous Discrete n n i 1 b i 1 b E k k P X i k P X i k since Continuous E k k f X dx k a n P X i 1 i 1 b f X dx since f X dx 1 a a Note that this means that practically every property of integration has an equivalent property in expected value 3 E f g E f E g Note that E f g where f and g are sets of outcomes withe same probability distribution n n n i 1 i 1 i 1 f g P X i f P X i g P X i The last two terms are the respective expected values The continuous model property is similar 4 Combining the two properties give a linear combination E k1 f k2 g k1E f k2 E g Right now you might give a big yawn and a who cares However we will use these properties in a moment after we define variance Variance Var X 2 Variance Var X is defined as follows for the respective discrete or continuous models n Discrete 2 2 Var X E X X i P X i i 1 b 2 2 Continuous Var X E X x f x dx a Var X is a measure of spread dispersion of data in the probability model It is the weighted total of all squares of distances from the mean to every point in the data More About Expected Value and Variance Page 2 of 5 However mathematicians dislike the idea of a measure We want the measure Hence we want the minimum total of distances between some optimum point and every point in the data set This is where our properties come in Let s assume there is some optimum point b then discover what it must be We also assume that there is a random variable X with an expected value E X 2 Then E X b is the weighted total of all squares of distances from b to every point in the data Using our properties and some algebra we get the result to the right Recalling that we want an optimum point so we optimize the functions with respect to b by taking 2 E X b 1 b We see that 2 E X b E X 2 2 Xb b 2 E X 2 E 2 Xb E b 2 E X 2 2bE X b 2 E X 2 2b b 2 2 E X b E X 2 2b b 2 2 2b since the mean is constant for the b b model From this point it is the same old game We solve 2 2b 0 to find that b We also observe or use the second derivative to prove that b is a minimum 2 So the optimum measure of spread is the value E X Done Our definition is the best To get a more practical formula for variance we use the same properties as shown to the right So what does Var X mean Considering that it is the weighted total of all squares of distances from the data to the mean a large relative value of Var X means distances are greater Hence the spread of data from the mean is great If Var X is small the mean is close to the preponderance of data points Hence the data is tightly grouped around the mean 1 Notice since 2 E X b 2 E X E X 2 2 X 2 E X 2 E 2 X E 2 E X 2 2 E X 2 E X 2 2 2 E X 2 2 is a function of two variables I chose to use the partial derivative More About Expected Value and Variance Standard Deviation Page 3 of 5 Var X Variance is not easy to analyze Hence we take the square root to find standard deviation Var X Now we can think about the weighted total distance deviation from the mean to a data point in terms of the standard value This is discussed in another lesson as the z score z x Some Formulas for Mean and Variance This is the single most mathematically demanding section of this course in many respects However at the end you will have a number of very nice formulas that will simplify your life By definition E X b n X i 1 i P X i for discrete models or E X x P x dx for continuous a models It is tedious to calculate for every standard model using these definitions so let s do it theoretically using standard assumptions Assume the model is uniform and discrete Hence P X n 1 Then P X i np 1 P X p n i 1 We talked about uniform discrete models a long time ago Formulas p is a constant Using the standard rule for the sum of consecutive integers 1 through n we get the following n n 1 1 n 1 1 n n E X X i P X i X i X i 2 n n i 1 n i 1 i 1 E X 1 n 2 We are struck with the fact that for a uniformly distributed list of consecutive integers the mean is the average of the first and last value if we start at x 1 Now watch our E X properties work for us Suppose we add k to each face of the die then E X k 1 n E k 1 n k 2 2 1 n 2k k 1 k n E X k 2 2 Now we have the fact that for a uniformly distributed list of consecutive integers the mean is the average of the first and last value regardless of where we start Applications a Roll a die and use the value on the face as the random variable E X 1 6 3 5 2 More About Expected Value and Variance b Page 4 of 5 Add 3 to each face value Then the expected value is E X 3 1 6 3 3 5 3 6 5 2 c Find the expected value of the integers 9 x 25 if they are uniformly distributed as random variables 9 25 1 17 17 or E X 8 8 9 8 17 2 2 Suppose the model is uniform and continuous on the interval a x b Then the probability density E X b function f X p …