Fundamentals of Probability and Statistics for Engineers

In comparison with Equation (5.7), Equation (5.10) yields a different relation- ship between the PDFs of X and Y owing to a diff ...

F ollowing Equation (5.12), the result is It is valid over the entire range as it is in correspondence with the range defined in ...

and The pdf of Y is thus given by The range space RY corresponding to is The pdf given above is thus valid for the whole range o ...

0.4 0.2 –2 2 4 y fY(y) –4 Figure 5. 6 Probability density function, fY (y) in Example 5.4 r 0 i VR + Figure 5.7 Circuit for ...

and The range 90 110 corresponds to 19 v 21 in the range space RV .Itis clear that fV(v) is zero outside the interval 19 v 21. I ...

In the examples given above, it is easy to verify that all density functions obtained satisfy the required properties. Let us no ...

Since derivative dg 21 (y) dy is negative whereas the others are positive, Equation (5.22) takes the convenient form Figure 5.11 ...

where r is the number of roots for x of equation y g(x). Clearly, Equation (5.12) is co ntained in this theorem as a sp ecial ca ...

For all y, Equation (5.25) yields a result identical to the solution for Example 5.4 [see Equation (5.18)]. Ex ample 5. 7. Probl ...

H ence, using Equation (5.25), or This is the so-called^2 distribution, to be discussed in more detail in Section 7.4.2. Example ...

where The theorems stated in this section do not apply in this case to the portions v 1 <95Vandv 1 > 105V because in finit ...

of Y if they exist. However, this procedure – the determination of moments of Y on finding the probability law of Y – is cumbers ...

Upon evaluating Y (t), the moments of Y are given by [Equation (4.52)]: Ex ample 5. 9. Problem: a random variable X is discrete ...

As expected, these answers agree with the results obtained earlier [Equations (5.34) and (5.35)]. Let us again remark that the p ...

determination of the distribution Y when all Xj, j 1,2,...,n, are continuous random variables. Consider the transformation where ...

Substituting fX 1 X 2 (x 1 ,x 2 ) into Equation (5.44) enables us to determine FY (y) and, on differentiating with respect to y, ...

As a numerical example, suppose that X 1 and X 2 are independent and The pdf of Y is, fo llowing Equation (5.45), In the above, ...

The region R^2 for positive values of x 1 and x 2 is shown as the shaded area in F igure 5.18. H ence, y fY(y) 01 1 2 2 Figure 5 ...

For independent X 1 and X 2 , The pdf of Y is thus given by, on differentiating Equation (5.49) with respect to y, and, on subst ...

Example 5.14.Problem: in structural reliability studies, the probability of failure q is defined by where R and S represent, res ...