Fundamentals of Probability and Statistics for Engineers

which is the desired solution. If random variables X 1 and X 2 are independent, straightforward differentiation shows that Let u ...

We have not given examples in which functions of more than two random variables are involved. Although more complicated problems ...

Considerable importance is attached to the results expressed by Equations (5.54) and (5.55) because sums of random variables occ ...

where the integration limits are determined from the requirements y x 2 >0, and x 2 > 0. The result is Let us note that th ...

The problem is to obtain the joint probability distribution of random variables Yj, j 1,2,...,m, which arise as functions of n j ...

where the integrals represent n-fold integrals with respect to the components of xandy, respectively. Following the usual rule o ...

where r is the number of solutions for x of equation yg( x), and Jj is defined by In the above, gj 1 ,gj 2 ,...,and gjn are comp ...

H ence, Equation (5.67) leads to It is of interest to note that the result given by Equation (5.72) can be written as where impl ...

Equation (5.68) now applies and we have where On substituting Equations (5.75) and (5.76) into Equation (5.74), we have We note ...

In order to conform with conditions stated in this section, we augment Equa- tion (5.78) by some simple transformation such as T ...

Problems 5.1 Determine the Probability distribution function (PDF) of Y 3 X 1 if (a) Case 1: (b) Case 2: 5.2 Temperature C measu ...

5.6 The fo llowing arises in the study of earthquake-resistant design. If X is the magni- tude of an earthquake and Y is ground- ...

5.10 At a given location, the PDF of annual wind speed, V, in miles per hour is found to be The wind force W exerted on structur ...

Suppose that voltage source v in the circuit is a deterministic constant. (a) Find the pdf of current I, where I v/R, passing th ...

5.20 Consider a system with a parallel arrangement, as shown in Figure 5.24, and let A be the primary co mponent and B its redun ...

where is a constant. Suppose that demand X 2 at this location during the same time interval has the same distribution as X 1 and ...

Determine the jpdf of Y 1 and Y 2 , defined by and show that they are independent. 5.29 The jpdf of X and Y is given by Determin ...

6 SOME IMPORTANT DISCRETE DISTRIBUTIONS This chapter deals with some distributions of discrete random variables that are importa ...

a sequence of Bernoulli trials can be symbolically represented by and, owing to independence, the probabilities of these possibl ...

where is the binomial coefficient in the binomial theorem In view of its similarity in appearance to the terms of the binomial t ...