ability distributions of S1 and S2 are fS1 1s1 2 and fS2 1s2 2 s2 , 8 0 s2 4 2s1, 0 s1 1

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where b is a constant that depends on the temperature of the gas and the mass of the particle. (a) Find the value of the constant a. (b) The kinetic energy of the particle is W mV 2 2 . Find the probability distribution of W. S5-8. Suppose that X has the probability distribution fX 1x2 1, 1 x 2

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Find the probability distribution of the random variable Y eX. S5-9. Prove that Equation S5-3 holds when y h(x) is a decreasing function of x. S5-10. The random variable X has the probability distribution fX 1x2 x 8, 0 x 4

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(a) Find the joint distribution of the area of the rectangle A S1 S2 and the random variable Y S1. (b) Find the probability distribution of the area A of the rectangle. S5-12. Suppose we have a simple electrical circuit in which Ohm s law V IR holds. We are interested in the probability distribution of the resistance R given that V and I are independent random variables with the following distributions: fV 1v2 and fI 1i2 1, 1 i 2 e v, v 0

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Find the probability distribution of Y (X 2)2. S5-11. Consider a rectangle with sides of length S1 and S2, where S1 and S2 are independent random variables. The prob-

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Find the probability distribution of R.

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MOMENT GENERATING FUNCTIONS (CD ONLY)

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Suppose that X is a random variable with mean . Throughout this book we have used the idea of the expected value of the random variable X, and in fact E(X) . Now suppose that we are interested in the expected value of a particular function of X, say, g(X) X r. The expected value of this function, or E[g(X)] E(X r), is called the rth moment about the origin of the random variable X, which we will denote by r. De nition The rth moment about the origin of the random variable X is E1X 2

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r r a x f 1x2, x

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X discrete (S5-7) X continuous

xr f 1x2 dx,

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Notice that the rst moment about the origin is just the mean, that is, E1X2 . 1 , we can write the variFurthermore, since the second moment about the origin is E1X 2 2 2 ance of a random variable in terms of origin moments as follows:

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E1X 2 2

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The moments of a random variable can often be determined directly from the de nition in Equation S5-7, but there is an alternative procedure that is frequently useful that makes use of a special function. De nition The moment generating function of the random variable X is the expected value of e tX and is denoted by MX (t). That is, MX 1t2 E1e 2

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X discrete (S5-8) X continuous

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etx f 1x2 dx,

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The moment generating function MX (t) will exist only if the sum or integral in the above definition converges. If the moment generating function of a random variable does exist, it can be used to obtain all the origin moments of the random variable.

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Let X be a random variable with moment generating function MX (t). Then r d r MX 1t2 ` dt r t (S5-9)

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Assuming that we can differentiate inside the summation and integral signs, d r MX 1t2 dt r

r tx a x e f 1x2, x

X discrete X continuous

x re tx f 1x2 dx,

Now if we set t

0 in this expression, we nd that d rMX 1t2 ` dt r t E1X r 2

EXAMPLE S5-5

Suppose that X has a binomial distribution, that is f 1x2 n a b px 11 x p2 n x, x 0, 1, p , n

Determine the moment generating function and use it to verify that the mean and variance of the binomial random variable are np and 2 np(1 p). From the de nition of a moment generating function, we have MX 1t2

tx n x a e a x b p 11 x 0 n

p2 n

n t x a a x b 1pe 2 11 x 0

p2 n

5-10

This last summation is the binomial expansion of [pet MX 1t2 dMX 1t2 dt 3pet 11 (1 p2 4 n p)]n, so

Taking the rst and second derivatives, we obtain M 1t2 X and M 1t2 X If we set t d 2MX 1t2 dt 2 npet 11 p npet 2 31 p1et 12 4 n

npet 31

p1et

12 4 n

0 in M 1t2 , we obtain X M 1t2 0 t X M 1t2 0 t X

np 0 in M 1t2, X

which is the mean of the binomial random variable X. Now if we set t

np11

Therefore, the variance of the binomial random variable is

np11

1np2 2

np11

EXAMPLE S5-6

Find the moment generating function of the normal random variable and use it to show that the mean and variance of this random variable are and 2, respectively. The moment generating function is MX 1t2 etx 1 12 1 12 e