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Expected Value of a Random Variable
The mean of a random variable, also known as its expected
value, is the weighted average of all the values that a
random variable would assume in the long run. The expected value
of a random variable can be thought of this way: the random
variable is made to assume values according to its probability
distribution, all the values are recorded and the mean is
computed. If this process is repeated indefinitely, the
calculated mean of the values will approach some finite quantity,
assuming that the mean of the random variable does exist (i.e.,
it does not diverge to infinity). This finite value is the mean
of the random variable.
The expected value of a random variable X is denoted by E(X).
For a discrete random variable, E(X) is calculated as
EX. The random variable X has the following probability
distribution:
| x |
pX(x) |
| |
|
| 2 |
1 / 36 |
| 3 |
2 / 36 |
| 4 |
3 / 36 |
| 5 |
4 / 36 |
| 6 |
5 / 36 |
| 7 |
6 / 36 |
| 8 |
5 / 36 |
| 9 |
4 / 36 |
| 10 |
3 / 36 |
| 11 |
2 / 36 |
| 12 |
1 / 36 |
The random variable X assumes a value equal to the sum of two
dice rolls. Its expected value is calculated as
= 2(1/36) + 3(2/36) + 4(3/36) + 5(4/36) + 6(5/36) + 7(6/36) +
8(5/36) + 9(4/36) + 10(3/36) + 11(2/36) + 12(1/36)
= (1/36) (2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12)
= (252/36) = 7
In the long run, the average value of two dice rolls using
regular dice is 7.
The expected value of the function g(X) of a discrete random
variable X is the mean of another random variable Y which assumes
the values of g(X) according to the probability distribution of
X. Denoted by E[g(X)], it is calculated as
The expected value of a discrete random variable X is actually
a special case of E[g(X)], where g(X) = x.
The expected value of the function g(X,Y) of two discrete
random variables, with joint probability mass function pX,Y(x,y), is denoted by E[g(X,Y)] and
is calculated as
This definition can be extended to three or more discrete
random variables. In general, if a set of random variables X1 , X2,
.... , Xn has a joint
probability function p(x1 , x2 , ...., xn),
the expected value of the function g(X1
, X2 , .... , Xn
) is given by
EX. The random variable X that assumes the value of a dice
roll has the probability mass function p(x) = 1/6 for x Î {1, 2, 3, 4, 5, 6}. If g(X) = X2, then
= (1/6) (1 + 4 + 9 + 16 + 25 + 36)
= 91/6
EX. Two random variables X and Y have the following joint
probability distribution
| (x,y) |
pX,Y(x,y) |
| (1,1) |
1/3 |
| (1,2) |
1/8 |
| (2,1) |
1/2 |
| (2,2) |
1/24 |
The expected value of the function g(X,Y) = XY is calculated
as follows:
= (1)(1)(1/3) + (1)(2)(1/8) + (2)(1)(1/2) + (2)(2)(1/24)
= 1/3 + 1/4 + 1 + 1/6
= 7/4
For any discrete random variable whose expected value exists
and is E(X), the expected value of g(X) = aX + b, where a and b
are constants, is given by
From this equality,
(1) E[aX] = aE(X)
(2) E[b] = b
The first statement asserts that the expected value of a
scalar function of a random variable is the product of the
expected value of the random variable and the scalar value. The
second statement asserts that the expected value of a constant is
the constant itself.
The expected value of the sum of two discrete random variables
X and Y is
Similarly, E(X - Y) = E(X) - E(Y).
The mean of the product of two independent discrete random
variables is
The mean of the product of two independent random variables is
simply the product of the mean of the two independent random
variables.
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