Applied Statistics and Probability for Engineers

5-8

S5-7. The velocity of a particle in a gas is a random variable

Vwith probability distribution

where bis 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. Find the

probability distribution of W.

S5-8. Suppose that Xhas the probability distribution

Find the probability distribution of the random variable YeX.

S5-9. Prove that Equation S5-3 holds when yh(x) is a

decreasing function of x.

S5-10. The random variable Xhas the probability distribution

Find the probability distribution of Y (X 2)^2.

S5-11. Consider a rectangle with sides of length S 1 and S 2 ,

where S 1 and S 2 are independent random variables. The prob-

fX 1 x 2 

x

8 ,^0 x^4

fX 1 x 2 1, 1 x 2

WmV (^2)  2
fV 1 v 2 av^2 ebv v 0
ability distributions of S 1 and S 2 are
and
(a) Find the joint distribution of the area of the rectangle A
S 1 S 2 and the random variable Y S 1.
(b) Find the probability distribution of the area Aof the rec-
tangle.
S5-12. Suppose we have a simple electrical circuit in
which Ohm’s law VIRholds. We are interested in the
probability distribution of the resistance Rgiven that Vand
Iare independent random variables with the following dis-
tributions:
and
Find the probability distribution of R.
fI 1 i 2 1, 1 i 2
fV 1 v 2 ev, v 0
fS 2 1 s 22 
s 2
8
, 0 s 2  4
fS 1 1 s 12  2 s 1 , 0 s 1  1
5-9 MOMENT GENERATING FUNCTIONS (CD ONLY)
Suppose that Xis 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-
terested in the expected value of a particular function of X, say, g(X)Xr. The expected value of
this function, or E[g(X)]E(Xr), is called the rth moment about the origin of the random variable
X, which we will denote by .¿r
The rth momentabout the originof the random variable Xis
r¿E 1 Xr 2  μ (S5-7)
a
x

xr f 1 x 2 , X discrete



   

xrf 1 x 2 dx, X continuous

Definition

Notice that the first moment about the origin is just the mean, that is,.

Furthermore, since the second moment about the origin is , we can write the vari-

ance of a random variable in terms of origin moments as follows:

2 E 1 X^22  3 E 1 X 242  2 ¿^2

E 1 X^22 ¿ 2

E 1 X 2 ¿ 1 

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