Fundamentals of Probability and Statistics for Engineers

[see Equation (2.26)], for three events A, B, and C, we have, in the case of three random variables X, Y, and Z, Hence, for the ...

Consider now the conditional mass function pX Y (xy). With Y y having happened, the situation is again similar to that for deter ...

However, H ence, where is a constant. The desired fR(r A) is then obtained from the above by differentiation. We obtain It can b ...

Problems 3.1 For each of the functions given below, determine constant a so that it possesses all the properties of a probabilit ...

3.3 For pX (x) and f (^) X (x) in Figure 3.19(a) and 3.19(b) respectively, sketch roughly in scale the corresponding PDF FX (x) ...

3.7 Let T denote the life (in months) of a light bulb and let (a) Plot fT (t) against t. (b) Derive FT (t) and plot FT (t) again ...

3.10 Fire can erupt at random at any point along a stretch of forest AB. The fire station is located as shown in Figure 3.22. De ...

3.13 For each of the joint probability mass functions (jpmf), pX Y (x,y), or joint prob- ability density functions (j pdf), fX Y ...

3.17 Let random variable X denote the time of failure in years of a system for which the PDF is FX (x). In terms of FX (x), dete ...

Assume that X and Y are independent. Determine P(X 25 Y > 8), the prob- ability that the next earthquake within 50 miles will ...

(iii) The outcome at the (k 1)th generation is dependent only on the class order at the kth generation and not on any generation ...

4 EXPECTATIONS AND MOMENTS While a probability distribution [FX (x), pX (x), or f (^) X(x)] co ntains a complete description of ...

Let us note some basic properties associated with the expectation operator. For any constant c and any functions g(X) and h(X) f ...

Example 4.1.Problem: From Example 3.9 (page64), determine the average number of cars turning west in a group of n cars. Answer: ...

Answer: the average value required in this example is a conditional mean of R given the event A. Although no formal definition i ...

Let us note that the mean life, is given by A point xi such that where is an arbitrarily small positive quantity, is called a mo ...

An important relation between the variance and simple moments is This can be shown by making use of Equations (4.3). We get fX(x ...

We note two other properties of the variance of a random variable X which can be similarly verified. They are: where c is any co ...

Now, and Proceeding as in Example (4.1), Thus, and Ex ample 4. 6. We again use Equation (4.8) to determine the variance of X def ...

Answer: for this problem, it is reasonable to expect that errors introduced in the making of a three-foot tape measure again are ...