Download 2.4 Random Variables and Expectation

2.4
Random Variables and Expectation
Example 2.4.1 Two fair dice (blue and red) are rolled. The sample space S is the set of
36 possible outcomes, each consisting of two scores between 1 and 6. Let X denote the total
score - the sum of the two scores. Then X associates a number (between 2 and 12) to each
outcome - i.e. X is a funtion from S to R. We have
X(1, 6) = 7, X(3, 4) = 7, X(4, 6) = 10, etc.
Definition 2.4.2 A random variable is a function from a sample space S to the set R of
real numbers.
This is a formal definition but in practice our random variables will sensibly associate
numbers to outcomes of experiments.
Examples of Random Variables :
1. X1 : Number of clubs in a randomly dealt poker hand (5 cards).
S : Set of possible five-card hands.
X1 (S) = {0, 1, 2, 3, 4, 5} - set of possible values of X1 .
2. Experiment : Continue tossing a coin until Heads (H) is obtained for the first time.
Sample space S = {H, T H, T T H, T T T H, . . . } - lists consisting of a nubmer of T
followed by one H. X2 : Number of tosses up to and including the first H.
So X2 (H) = 1, X2 (T H) = 2, X2 (T T H) = 3, etc.
X2 (S) = {1, 2, 3, . . . } - the set of natural numbers.
3. Experiment : Select two points of R2 .
Sample Space S : Set of pairs of points.
X3 : Distance between the two points.
X3 (S) − [0, ∞) : the set of non-negative real numbers.
Note: X1 and X2 above are examples of discrete random variables; their sets of possible
values are “spaced out” along the real number line. X 3 is a continuous random variable its set of possible values is a continuous chunk of the real number line and is uncountable.
For now we will restrict our attention to discrete random variables.
Definition 2.4.3 Let X : S −→ R be a discrete random variable associated to an experiment
whose sample space is S. The probability function
pX : X(S) −→ [0, 1]
is defined by
pX (x) = P (X = x), for x ∈ X(S).
It is the function that associates to each possible value of X the probability that X will
assume that value in a performance of the experiment.
Example 2.4.4 Let X be the total score when two fair dice are rolled. Then X(S) =
{2, 3, . . . , 12} and
pX (2)
pX (3)
pX (4)
pX (5)
pX (6)
pX (7)
Note
12
k=2 pX (k)
=
=
=
=
=
pX (12)
pX (11)
pX (10)
pX (9)
pX (8)
=
=
=
=
=
=
= 1 as we would expect.
9
1/36
1/18
1/12
1/9
5/36
1/6
(= 2/36)
(= 3/36)
(= 4/36)
(= 6/36)
Example 2.4.5 An unfair coin will show H (heads) with probability 0.3 and T (tails) with
probability 0.7. Let X be the number of tosses until the first H. Then X(S) = {1, 2, . . . }
and
pX (1) =
pX (2) =
pX (3) =
P (X = 1) = P (H)
=
P (X = 2) = P (T H)
=
P (X = 3) = P (T T H) =
0.3
0.7 × 0.3
(0.7)2 × 0.3
In general for k ∈ X(S)
pX (k) = P (T
. . . T} H) = (0.7)k−1 (0.3).
| T {z
k−1
Note:
∞
X
pX (k) = 0.3 + (0.7)(0.3) + (0.7)2 (0.3) + . . .
k=1
- a geometric series with initial term 0.3 and common ratio 0.7. The sum is
0.3
= 1,
1 − 0.7
as expected.
For any discrete random variable X we must have
X
pX (x) = 1.
x∈X(S)
Definition 2.4.6 Let X : S −→ R be a discrete random variable. The expectation or
expected value of X, denoted by E(X) or sometimes µ, is defined by
X
E(X) =
xpX (x),
x∈X(S)
i.e. E(X) is the sum over all the possible values x of X of the expression xP (X = x).
Notes:
1. E(X) is a weighted average of the values of X, the weight assigned to each value being
its probability of occurring.
2. If X(S) is finite and all values of X are equally likely to occur, then E(X) is just
the average of these values. For example if X is the score on a fair die, then X(S) =
{1, 2, 3, 4, 5, 6} and
E(X) = 1 ×
1
1
1
1
1
1
1
+ 2 × + 3 × + 4 × + 5 × + 6 × = 21 × = 3.5
6
6
6
6
6
6
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3. E(X) may not be a possible value of X at all (as in the example above). However
E(X) is the value that x takes “on average” over a large number of repetitions of the
experiment.
Jistificiation for the Formula:
Suppose that X has k possible values x1 , x2 , . . . , xk and that
P (X = xi ) = pX (i) = pi ,
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for i = 1, . . . , k. So pi is the probability that X takes the value xi . Suppose that in n
reperitions of the experiment, the number of times that X takes the value x i is ni . Then
the sum of the values of X over the n repetitions is
n1 x 1 + n 2 x 2 + · · · + n k x k ,
and the average value of X is
n1
n2
nk
n1 x 1 + n 2 x 2 + · · · + n k x k
=
x1 +
x2 + · · · +
xk .
n
n
n
n
Now nni is the proportion of the n repetitions in which X = x i . As n gets large we expect
this number to approach pi , so we expect the average value of X to approach
x1 p1 + x2 p2 + · · · + xk pk = E(X).
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