1. MathematicalMarketing Slide 4a.1 Distributions Chapter 4: Part a – The Expectation and Variance of Distributions We will be discussing  The Algebra of Expectation  The Algebra of Variance  The Normal Distribution (These topics are needed for Chapter 5)

2. MathematicalMarketing Slide 4a.2 Distributions The Expectation of a Discrete Random Variable Assume we have a random variable, a i. If a i can take on only certain values: 1, 2, ···, J, then we have as the definition of Expectation of a i, or E(a i )

3. MathematicalMarketing Slide 4a.3 Distributions The Expectation of a Continuous Variable By definition, the expectation of that scalar is Imagine we have a scalar, a i, that can take on any possible values with probability f(a i ), i. e. f(a i ) aiai

4. MathematicalMarketing Slide 4a.4 Distributions Three Rules for E(·) The Expectation of a Sum is the Sum of the Expectations E(a + b) = E(a) + E(b) The Expectation of a Constant is That Constant E(c) = c In the Expectation of a Linear Combination, a Constant Matrix Can Pass Through E(·) E(Da) = DE(a) E(a′F) = E(a′)F

5. MathematicalMarketing Slide 4a.5 Distributions The Variance of a Random Variable The Variance of the Random Vector a is Given by

6. MathematicalMarketing Slide 4a.6 Distributions The Variance of a Mean Centered Vector If E(a) = 0, i. e. a is mean centered, we have just V(a) = E(aa′) NB – just because E(a) = 0 doesn’t mean that E(aa′) = 0!

7. MathematicalMarketing Slide 4a.7 Distributions The Variance Matrix For mean centered a, we have V(a) = E(aa′)

8. MathematicalMarketing Slide 4a.8 Distributions Two Rules for V(·) Adding a Constant Vector Does Not Change the Variance V(a + c) = V(a) The Variance of a Linear Combination is a Quadratic Form V(Da) = DV(a)D′ Hint: The Above Theorem Will Figure Many Many Times in What Is to Come!

9. MathematicalMarketing Slide 4a.9 Distributions The Normal Density Function xaxa  x Pr(x) 1.0 0 Consider a random scalar x. Under the normal distribution, the probability that x takes on the value x a is given by the equation

10. MathematicalMarketing Slide 4a.10 Distributions The Standard Normal Density If  = 0 and  2 = 1, so that z = (x -  ) /  the expression simplifies to Note very common notation

11. MathematicalMarketing Slide 4a.11 Distributions The Normal Distribution Function xbxb x Pr(x) 1.0 0 According to the normal distribution function, the probability that the random scalar x is less than or equal to some value x b is

12. MathematicalMarketing Slide 4a.12 Distributions Other Notational Conventions For our scalar x that is distributed according to the normal distribution function, we say x ~ N( ,  2 ).

13. MathematicalMarketing Slide 4a.13 Distributions The Standardized Normal Distribution If we set z = (x -  ) /  then Again, note the notational convention and that

14. MathematicalMarketing Slide 4a.14 Distributions The Normal Ogive  (z) 1.0 0 zbzb