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Chapter 2 Discrete Random Variables and Probability Distributions
9. The random variable Y has the probability distribution
g.y = 1
4 if y = 2; 3; 4; or 5:
Find G.y, the distribution function for Y .
10. Find the distribution function for the geometric distribution f .x =
1 2
x
, x = 1; 2; 3 : : : :
11. A random variable, X, has the distribution function
F.x = 8
: 0; x −1
1 3
; −1 ≤ x 0
5 6
; 0 ≤ x 2
1; x ≥ 2 :
Find the probability distribution function, f .x:
12. A random variable X is defined on the integers 0; 1; 2; 3; : : : ; and has distribution
function F.x. Find expressions, in terms of F.x, for
a. P.a X b b. P.a ≤ X b
c. P.a X ≤ b d. P.a ≤ X ≤ b
13. If f .x =
1 n
; x = 1; 2; 3; : : : ; n so that each value of X has the same probability
then X is called a discrete uniform random variable. Find the distribution function for this random variable.
2.3
c
Expected Values of Discrete Random Variables
2.3.1 Expected Value of a Discrete Random Variable
Random variables are easily distinguished by their probability distribution functions. They are also often characterized or described by measures that summarize these
distributions. Usually, “average” values, or measures of centrality, and some measure of their dispersion, or variability, are found as values characteristic of the distribution.
We begin with the definition of an average value for a discrete random variable, X, denoted by E. X, or ¼
x
; which we will call the expectation, or expected value, or
mean, or mean value all of these terms are in common usage of X.
2.3 Expected Values of Discrete Random Variables
89 Definition:
E. X = ¼
x
= X
x
x · P.X = x; provided the sum converges, where the summation occurs over all the discrete values
of the random variable, X. Note that each value of the random variable X is weighted by its probability in the sum.
The provision that the sum be convergent cautions us that the sum may, indeed, be infinite. There are random variables, otherwise seemingly well-behaved, that have no
mean value. This definition is, in reality, a simple extension of what the reader would recognize
as an average value. Consider an example.
Ex ample 2.3.1.1
A student has examination grades of 82, 91, 79, and 96 in a course in probability. We would no doubt calculate the average grade as
82 + 91 + 79 + 96 4
= 87: This could also be calculated as
82 · 1
4 + 91 ·
1 4
+ 79 · 1
4 + 96 ·
1 4
= 87; where the examination scores have now been equally weighted. Should the
instructor decide to weight the fourth examination three times as much as any one of the other examinations, this simply changes the weights and the
average examination grade is then
82 · 1
6 + 91 ·
1 6
+ 79 · 1
6 + 96 ·
3 6
= 90: So the idea of adding scores multiplied by their probabilities is not a
new one. This is exactly what we do when we calculate E. X:
b
Ex ample 2.3.1.2
If a fair die is thrown once, as in Example 2.1.1, the average result is ¼
x
= 1 · 1
6 + 2 ·
1 6
+ 3 · 1
6 + 4 ·
1 6
+ 5 · 1
6 + 6 ·
1 6
= 7
2 :
So we recognize
7 2
, or 3.5, as the average result, although 3.5 is not a possible value for the face showing on the die. What is the meaning of
this? The interpretation is as follows: If we threw a fair die a large number
90
Chapter 2 Discrete Random Variables and Probability Distributions
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