4 1 x dx 3 2

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Note that if X has certain dimensions or units, such as centimeters (cm), then the variance of X has units cm2 while the standard deviation has the same unit as X, i.e., cm. It is for this reason that the standard deviation is often used.

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Some Theorems on Variance

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Theorem 3-4 where m Theorem 3-5 E(X). s2 E[(X m)2] E(X2) m2 E(X2) [E(X)]2 (16)

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If c is any constant, Var(cX) c2 Var(X) m E(X). (17)

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Theorem 3-6 The quantity E[(X

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a)2] is a minimum when a

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Theorem 3-7 If X and Y are independent random variables, Var (X Var (X Y) Y) Var (X) Var (X) Var (Y) Var (Y) or or s2 X

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s2 X

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s2 Y s2 X s2 Y

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(18) (19)

s2 X

Generalizations of Theorem 3-7 to more than two independent variables are easily made. In words, the variance of a sum of independent variables equals the sum of their variances.

Standardized Random Variables

Let X be a random variable with mean m and standard deviation s (s dardized random variable given by X* X s m 0). Then we can define an associated stan-

(20)

An important property of X* is that it has a mean of zero and a variance of 1, which accounts for the name standardized, i.e., E(X*) 0, Var(X*) 1 (21)

The values of a standardized variable are sometimes called standard scores, and X is then said to be expressed in standard units (i.e., s is taken as the unit in measuring X m). Standardized variables are useful for comparing different distributions.

Moments

The rth moment of a random variable X about the mean m, also called the rth central moment, is defined as mr E [(X m)r] (22)

CHAPTER 3 Mathematical Expectation

where r 0, 1, 2, . . . . It follows that m0 1, m1 0, and m2 s2, i.e., the second central moment or second moment about the mean is the variance. We have, assuming absolute convergence, mr

a (x 3 `(x

m)r f(x) m)r f(x) dx

(discrete variable) (continuous variable)

(23) (24)

The rth moment of X about the origin, also called the rth raw moment, is defined as mr r E(Xr) 0. (25)

where r 0, 1, 2, . . . , and in this case there are formulas analogous to (23) and (24) in which m The relationship between these moments is given by mr mr r r mr 1 m 1 r m2 m3 m4 c 1, r ( 1) j mr j m j j r 2m3 6mr m2 2 c ( 1)rmr mr 0

(26)

As special cases we have, using mr 1

m and mr 0 mr 2 mr 3 mr 4

m2 3mr m 2 4mr m 3

(27) 3m4

Moment Generating Functions

The moment generating function of X is defined by MX (t) that is, assuming convergence, MX(t)

E(etX )

(28)

a etx f (x)

tx 3 `e f(x) dx

(discrete variable)

(29)

MX (t)

(continuous variable)

(30)

We can show that the Taylor series expansion is [Problem 3.15(a)] MX (t) 1 mt mr 2 t2 2! c mr r tr r! c (31)

Since the coefficients in this expansion enable us to find the moments, the reason for the name moment generating function is apparent. From the expansion we can show that [Problem 3.15(b)] mr r i.e., mr is the rth derivative of MX (t) evaluated at t r stead of MX (t). dr M (t) 2 dtr X t (32)

0. Where no confusion can result, we often write M(t) in-

Some Theorems on Moment Generating Functions

Theorem 3-8 If MX (t) is the moment generating function of the random variable X and a and b (b 2 0) are constants, then the moment generating function of (X a) > b is M(X

a)>b(t)

t eat>bMX b

(33)

CHAPTER 3 Mathematical Expectation

Theorem 3-9 If X and Y are independent random variables having moment generating functions MX (t) and MY (t), respectively, then MX

Y (t)

MX (t) MY (t)

(34)

Generalizations of Theorem 3-9 to more than two independent random variables are easily made. In words, the moment generating function of a sum of independent random variables is equal to the product of their moment generating functions. Theorem 3-10 (Uniqueness Theorem) Suppose that X and Y are random variables having moment generating functions MX (t) and MY (t), respectively. Then X and Y have the same probability distribution if and only if MX (t) MY (t) identically.