3.3 Expected Values

Contents

• 3.3.1 The Expected Value of X
• 3.3.2 The Expected Value of a Function
• 3.3.3 Expected Value of a Linear Function
• 3.3.4 The Variance of X
• 3.3.5 A shortcut formula for variance
• 3.3.6 Variance of a Linear Function
• EXERCISES Section 3.3 (29-45)

Introduction

• University with 15,000 students
• Let $X$ represent the number of courses a randomly selected student is registered for
  • The pmf (probability mass function) of $X$ is provided
    • Example: $p(1)=0.01$
      • Calculation: $(0.01)\cdot 15,000=150$
        • 150 students are registered for one course
    • Similarly for other $x$ values:
      • $p(x)\cdot 15,000$ gives the number of students registered for $x$ courses

(3.6)

$X$	1	2	3	4	5	6	7
$p(X)$	.01	.03	.13	.25	.39	.17	.02
Number registered	150	450	1950	3750	5850	2550	300

• The average number of courses per student (average value of $X$) is calculated as:

  • Total number of courses taken by all students divided by the total number of students

• Example breakdown: - 150 students taking 1 course contribute $1(150)=150$ courses to the total - 450 students taking 2 courses contribute $2(450)=900$ courses to the total - And so on for other groups of students

  • The population average value of $X$ is given by: \frac{1(150) + 2(450) + 3(1950) + \cdots + 7(300)}{15,000} = 4.57 \tag{3.7}

• Alternative expression:

  • Since:
    • $\frac{150}{15,000}=0.01=p(1)$
    • $\frac{450}{15,000}=0.03=p(2)$
    • And similarly for other values
  • The population average value of $X$ can also be expressed as: 1 \cdot p(1) + 2 \cdot p(2) + \cdots + 7 \cdot p(7) \tag{3.8}
  • This alternative expression shows that the average value of $X$ is a weighted sum of the number of courses, weighted by their probabilities.

• Expression (3.8) shows:

  • To compute the population average value of $X$, we only need:
    • The possible values of $X$
    • Their corresponding probabilities (proportions)
  • The population size is irrelevant as long as the pmf is given by (3.6)

• The average or mean value of $X$:

  • Is a weighted average of the possible values $1,\dots ,7$
  • The weights are the probabilities of those values
  • For example, $p(1)$, $p(2)$, …, $p(7)$

• This simplifies the computation since the exact number of students isn’t necessary;

  • we just need the distribution of the number of courses each student takes.