Fundamentals of Probability and Statistics for Engineers

Further Reading Clark, R.D., 1946, ‘‘An Application of the Poisson Distribution’’, J. Inst. Actuaries 72 48–52. Solloway, C.B., ...

Problems 6.1 The random variable X has a binomial distribution with parameters (n,p). Using the formulation given by Equation (6 ...

6.10 A manufacturing firm receives a lot of 100 parts, of which 5 are defective. Suppose that the firm accepts all 100 parts if ...

6.19 For Example 6.10, determine the jpmf of X 1 and X 2. Determine the probability that, of the 10 persons studied, fewer than ...

6.26 Each air traffic controller at an airport is given the responsibility of monitoring at most 20 takeoffs and landings per ho ...

6.32 Suppose that , the mean rate of arrival, in the Poisson distribution is time- dependent and is given by Determine pmf the p ...

...

7 SOME IMPORTANT CONTINUOUS DISTRIBUTIONS Let us turn our attention to some important continuous probability distribu- tions. Ph ...

which is graphically presented in Figure 7.1(b). The mean, mX, and variance, , of X are easily found to be The uniform distribut ...

We are interested in the probability P( 3 X 0). As seen from Fi gure 7.2, it is clear that this probability is equal to the rati ...

It takes the shape of a flat surface bounded by (a 1 ,b 1 ) along the x axis and (a 2 ,b 2 ) along the y axis. We have seen an a ...

To answer the first question, in part (a), we integrate fX Y (x, y) over an appropriate region in the (x,y) plane satisfying y x ...

It is noteworthy that, if the independence assumption is removed, the jpdf of two uniformly distributed random variables will no ...

Its corresponding PDF is which cannot be expressed in closed form analytically but can be numerically evaluated for any x. The p ...

graph of fX (x) in this particular case is the well-known bell-shaped curve, symmetrical about the origin [Figure 7.6(a)]. Let u ...

Let us note in passing that 2 , the coefficient of excess, defined by Equation (4.12), for a normal distribution is zero. H ence ...

Expanding in a MacLaurin series as indicated by Equation (4.49), we can write In the last step we have used the elementary ident ...

of no surprise. As the number of steps increases, it is expected that position of the particle becomes normally distributed in t ...

Theorem 7. 2: Let X be a normal random variable with distribution N(m,^2 ). Then (X m is the standardized normal random variable ...

and 11.2 cm, what proportion of the manufactured parts will be rejected on average? Answer: If X is used to denote the part leng ...