Fundamentals of Probability and Statistics for Engineers

and therefore which gives the desired inversion formula. In summary, the transform pairs given by Equations (4.46), (4.47), (4.5 ...

Then, by definition, Since X 1 ,X 2 ,...,Xn are mutually independent, Equation (4.36) leads to We thus have which was to be prov ...

The distribution given by Equation (4.72) is called a gamma distribution, which will be discussed extensively in Section 7.4. Ex ...

Letting k 2 i n, we get Comparing Equation (4.76) with the definition in Equation (4.46) yields the mass function Note that, if ...

Our derivation of Equation 4.81 has been purely analytical. In his theory of Brownian motion, Einstein also obtained this result ...

If random variables X and Y are independent, then we also have To show the above, we simply substitute fX(x)f (y) for f (^) XY ( ...

They are, respectively, the position of the particle after 2 n steps and its position after 3 n steps relative to where it was a ...

Hence, as n Now, substituting Equation (4.94) into Equation (4.87) gives which can be evaluated following a change of variables ...

Further Reading and Comments As mentioned in Section 4.2, the Chebyshev inequality can be improved upon if some additional distr ...

random variable representing distance of the hit from the center. Suppose that the pdf of R is Compute the mean score of each sh ...

4.9 The diameter of an electronic cable, say X, is random, with pdf (a) What is the mean value of the diameter? (b) What is the ...

times equally likely to choose either direction, determine the average time interval (in minutes) that the miner will be trapped ...

(a) Determine mV and^2 V of voltage V, which is given by (b) Determine the correlation co efficient of R and V. 4.23 Let the jpd ...

4.30 Determine the characteristic function corresponding to each of the PDFs given in Problem 3.1(a)–3.1(e) (page 67). U se it t ...

...

5 FUNCTIONS OF RANDOM VARIABLES The basic topic to be discussed in this chapter is one of determining the relation- ship between ...

where g(X) is assumed to be a continuous function of X. Given the probability distribution of X in terms of its probability dist ...

5.1.1.1 Discrete Random Variables Let us fir st dispose of the case when X is a discrete random variable, since it requires only ...

5.1.1.2 Continuous Random Variables A more frequently en countered case ar ises when X is continuous with known PDF, FX(x), or p ...

is the inverse function of g(x), or the solution for x in Equation (5.5) in terms of y. Hence, Equation (5.7) gives the relation ...