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CHAPTER 3 Mathematical Expectation
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Table 3-1 xj f (xj) 0 1>6 20 1>6 0 1>6 40 1>6 0 1>6 30 1> 6
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It follows that the player can expect to win $5. In a fair game, therefore, the player should be expected to pay $5 in order to play the game.
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EXAMPLE 3.2 The density function of a random variable X is given by
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f (x) The expected value of X is then
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x e2 0
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0 x 2 otherwise
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E(X)
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3 `xf (x) dx
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Functions of Random Variables
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2 1 30x 2 x dx
2 2 x 30 2 dx
x3 2 2 6 0
Let X be a discrete random variable with probability function f (x). Then Y able, and the probability function of Y is h(y) P(Y y) a P(X x) a
g(X) is also a discrete random vari-
f(x) n), then y1h(y1) y2h(y2) c
5xZg(x) y6
5xZg(x) y6
If X takes on the values x1, x2, c , xn, and Y the values y1, y2, c , ym (m ymh(ym ) g(x1)f (x1) g(x2) f (x2) c g(xn)f(xn ). Therefore, E[g(X)] g(x1) f (x1)
g(x2)f(x2)
g(xn)f(xn ) (5)
a g(xj) f(xj)
a g(x)f(x)
Similarly, if X is a continuous random variable having probability density f(x), then it can be shown that
E[g(X)]
3 `g(x)f(x) dx
Note that (5) and (6) do not involve, respectively, the probability function and the probability density function of Y g(X). Generalizations are easily made to functions of two or more random variables. For example, if X and Y are two continuous random variables having joint density function f(x, y), then the expectation of g(X, Y) is given by
` ` `
E[g(X, Y)]
3 `g(x, y) f(x, y) dx dy
1 2x) x dx 2
EXAMPLE 3.3 If X is the random variable of Example 3.2,
E(3X2
2 3 `(3x
2x) f (x) dx
30 (3x
10 3
Some Theorems on Expectation
Theorem 3-1 If c is any constant, then E(cX) cE(X)
CHAPTER 3 Mathematical Expectation
Theorem 3-2 If X and Y are any random variables, then E(X Y) E(X) E(Y)
Theorem 3-3 If X and Y are independent random variables, then E(XY) Generalizations of these theorems are easily made. E(X)E(Y ) (10)
The Variance and Standard Deviation
We have already noted on page 75 that the expectation of a random variable X is often called the mean and is denoted by m. Another quantity of great importance in probability and statistics is called the variance and is defined by Var(X) E[(X m)2] (11)
The variance is a nonnegative number. The positive square root of the variance is called the standard deviation and is given by sX 2Var (X) 2E[(X m)2] (12)
Where no confusion can result, the standard deviation is often denoted by s instead of sX, and the variance in such case is s2. If X is a discrete random variable taking the values x1, x2, . . . , xn and having probability function f (x), then the variance is given by
s2 X
E[(X
m)2]
a (xj
m)2 f(xj)
a (x
m)2 f(x)
(13)
In the special case of (13) where the probabilities are all equal, we have s2 [(x1 m)2 (x2 m)2 c (xn m)2]>n (14)
which is the variance for a set of n numbers x1, . . . , xn. If X takes on an infinite number of values x1, x2, . . . , then s2 g ` 1 (xj m)2 f(xj), provided that the series j X converges. If X is a continuous random variable having density function f(x), then the variance is given by
s2 X
E[(X
m)2]
3 `(x
m)2 f(x) dx
(15)
provided that the integral converges. The variance (or the standard deviation) is a measure of the dispersion, or scatter, of the values of the random variable about the mean m. If the values tend to be concentrated near the mean, the variance is small; while if the values tend to be distributed far from the mean, the variance is large. The situation is indicated graphically in Fig. 3-1 for the case of two continuous distributions having the same mean m.
Fig. 3-1
CHAPTER 3 Mathematical Expectation
EXAMPLE 3.4 Find the variance and standard deviation of the random variable of Example 3.2. As found in Example 3.2, the mean is m E(X) 4 > 3. Then the variance is given by
and so the standard deviation is s
E B X
4 R 3
2 A9
22 3
4 f (x) dx 3
30 x
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