Fundamentals of Probability and Statistics for Engineers

which the value at Hence, is itself a random variable, and one of its very useful properties is that If Y is a discrete random v ...

It states that, in order to determine it can be found by taking a weighted average of the conditional expectation of X given eac ...

4.2 Chebyshev Inequality In the discussion of expectations and moments, there are two aspects to be considered in applications. ...

or In words, the probability of a three-foot tape measure being in error less than or equal to 0 06 feet is at least 0.75. Vario ...

here, the means of X and Y are, respectively, 10 and 01. Using Equation (4.19), for example, we obtain: where fX(x) is the margi ...

P roof of P ropert y 4. 2: to show Property 4.2, let t and u be any real quantities and form Since the expectation of a nonnegat ...

This result leads immediately to an important generalization. Consider a function of X and Y in the form g(X)h(Y) for which an e ...

correlation coefficient can vanish when the values of one random variable are completely determined by the values of another. Ex ...

and, from Equations (4.23) and (4.24), This is a simple example showing that X and Y are uncorrelated but they are completely de ...

the random column vector with components X 1 ,...,Xn, and let the means of X 1 ,...,Xn be represented by the vector mX. A conven ...

Verifications of these results are carried out for the case where X 1 ,...,Xn are continuous. The same procedures can be used wh ...

Let us verify Result 4.3 for n 2. The proof for the case of n random variables follows at once by mathematical induction. Consid ...

Ex ample 4. 11. Problem: an in sp ection is made of a group of n television picture tubes. If each passes the inspection with pr ...

tends to zero. In other words, random variable Y/n approaches the true mean with probability 1. Answer: to proceed with theproof ...

exhibits a random variation measured by v, then a physical process resulting from additive actions of n molecules will possess a ...

One of the important uses of ch aracteristic functions is in the determination of the moments of a random variable. Expanding X ...

Answer: according to Equation (4.46), U sing Equation (4.52), we have and The results for the mean and variance are the same as ...

Another useful expansion is the power series representation of the logarithm of the characteristic function; that is, where coef ...

Fourier transforms that fX (x) is uniquely determined from Equation (4.58); that is, no two distinct density functions can have ...

However, notice that, since X is continuous, P(X x) 0 if x is a point of continuity in the distribution of X. Hence, using Equat ...