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Conditional Expectation, Variance, and Moments
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If X and Y have joint density function f (x, y), then as we have seen in 2, the conditional density function of Y given X is f ( y u x) f (x, y) > f1 (x) where f1 (x) is the marginal density function of X. We can define the conditional expectation, or conditional mean, of Y given X by E(Y u X where X x is to be interpreted as x for conditional expectation. We note the following properties: 1. E(Y u X 2. E(Y) x)
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3 `y f(y u x) dy
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dx in the continuous case. Theorems 3-1 and 3-2 also hold
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E(Y) when X and Y are independent. x) f1(x) dx.
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It is often convenient to calculate expectations by use of Property 2, rather than directly.
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EXAMPLE 3.5 The average travel time to a distant city is c hours by car or b hours by bus. A woman cannot decide whether to drive or take the bus, so she tosses a coin. What is her expected travel time Here we are dealing with the joint distribution of the outcome of the toss, X, and the travel time, Y, where Y Ycar if X 0 and Y Ybus if X 1. Presumably, both Ycar and Ybus are independent of X, so that by Property 1 above
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E(Y u X and
E(Ycar u X l)
E(Ycar) 1)
E(Y u X
E(Ybus u X
E(Ybus)
Then Property 2 (with the integral replaced by a sum) gives, for a fair coin, E(Y) E(Y u X 0)P(X 0) E(Y u X 1)P(X 1) c 2 b
In a similar manner we can define the conditional variance of Y given X as E[(Y where m2 E(Y u X m2)2 u X
3 `(y
m2)2 f(y u x) dy
(56)
x). Also we can define the rth conditional moment of Y about any value a given X as E[(Y a)r u X x] 3 `(y a)r f (y u x) dy (57)
The usual theorems for variance and moments extend to conditional variance and moments.
CHAPTER 3 Mathematical Expectation
Chebyshev s Inequality
An important theorem in probability and statistics that reveals a general property of discrete or continuous random variables having finite mean and variance is known under the name of Chebyshev s inequality. Theorem 3-18 (Chebyshev s Inequality) Suppose that X is a random variable (discrete or continuous) having mean m and variance s2, which are finite. Then if P is any positive number, P(uX or, with P ks, P(uX mu ks) 1 k2 (59) mu P) s2 P2 (58)
EXAMPLE 3.6
Letting k
2 in Chebyshev s inequality (59), we see that P (u X mu 2s) 0.25 or P( u X mu 2s) 0.75
In words, the probability of X differing from its mean by more than 2 standard deviations is less than or equal to 0.25; equivalently, the probability that X will lie within 2 standard deviations of its mean is greater than or equal to 0.75. This is quite remarkable in view of the fact that we have not even specified the probability distribution of X.
Law of Large Numbers
The following theorem, called the law of large numbers, is an interesting consequence of Chebyshev s inequality. Theorem 3-19 (Law of Large Numbers): Let X1, X2, . . . , Xn be mutually independent random variables (discrete or continuous), each having finite mean m and variance s2. Then if Sn X1 X2 c ), Xn(n 1, 2, c Sn lim P 2 n m2 P 0 (60)
Since Sn > n is the arithmetic mean of X1, . . . , Xn, this theorem states that the probability of the arithmetic mean Sn > n differing from its expected value m by more than P approaches zero as n S ` . A stronger result, which we might expect to be true, is that lim Sn >n m, but this is actually false. However, we can prove that nS` lim Sn >n m with probability one. This result is often called the strong law of large numbers, and, by contrast, nS` that of Theorem 3-19 is called the weak law of large numbers. When the law of large numbers is referred to without qualification, the weak law is implied.
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