Page 220 - Applied Statistics And Probability For Engineers

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5-8

S5-7. The velocity of a particle in a gas is a random variable ability distributions of S 1 and S 2 are
V with probability distribution
1s 1 2 2s 1 , 0 s 1 1
f S 1
2 bv
f V 1v2 av e v 0
and
where b is a constant that depends on the temperature of the s 2
gas and the mass of the particle. f S 2 1s 2 2 , 0 s 2 4
8
(a) Find the value of the constant a.
2
(b) The kinetic energy of the particle is W mV 2 . Find the
probability distribution of W. (a) Find the joint distribution of the area of the rectangle A
S 1 S 2 and the random variable Y S 1 .
S5-8. Suppose that X has the probability distribution
(b) Find the probability distribution of the area A of the rec-
tangle.
f X 1x2 1, 1 x 2
S5-12. Suppose we have a simple electrical circuit in
X
Find the probability distribution of the random variable Y e . which Ohm’s law V IR holds. We are interested in the
probability distribution of the resistance R given that V and
S5-9. Prove that Equation S5-3 holds when y h(x) is a
I are independent random variables with the following dis-
decreasing function of x.
tributions:
S5-10. The random variable X has the probability distribution
v
f V 1v2 e , v 0
x
f X 1x2 , 0 x 4
8 and
f I 1i2 1, 1 i 2
2
Find the probability distribution of Y (X 2) .
S5-11. Consider a rectangle with sides of length S 1 and S 2 , Find the probability distribution of R.
where S 1 and S 2 are independent random variables. The prob-
5-9 MOMENT GENERATING FUNCTIONS (CD ONLY)

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 in-
r
terested in the expected value of a particular function of X, say, g(X) X . The expected value of
r
this function, or E[g(X)] E(X ), is called the rth moment about the origin of the random variable

X, which we will denote by .
¿
r
Definition
The rth moment about the origin of the random variable X is
r
x f 1x2, X discrete
a
x
r
E1X 2 (S5-7)
¿ r µ
r
x f 1x2 dx, X continuous

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

E1X 2 3E1X24 ¿ 2
2