Introduction to Probability and Statistics for Engineers and Scientists
164 Chapter 5: Special Random Variables For an illustration of the use of random numbers, suppose that a medical center is plann ...
5.4The Uniform Random Variable 165 each of the subsets of sizek−1 of the numbers 2,...,nis equally likely to be the other elemen ...
166 Chapter 5: Special Random Variables U 1 < .4 U 1 > .4 U 2 < .25 U 2 > .25 U 2 < .5 U 2 > .5 U 3 < –^13 ...
5.4The Uniform Random Variable 167 Because 1 = ∫ R f(x,y)dx dy = ∫ R cdxdy =c×Area ofR it follows that c= 1 Area ofR For any reg ...
168 Chapter 5: Special Random Variables − 3 03 (a) f(x) = 21 pe−x^2 /2 (b) 0.399 s m − 3s m − s mm + s m + 3s FIGURE 5.7 The nor ...
5.5Normal Random Variables 169 The moment generating function of a normal random variable with parametersμand σ^2 is derived as ...
170 Chapter 5: Special Random Variables An important fact about normal random variables is that ifXis normal with meanμ and vari ...
5.5Normal Random Variables 171 −x 0 x P {Z < −x} PP {Z > x} FIGURE 5.8 Standard normal probabilities. It remains for us to ...
172 Chapter 5: Special Random Variables =P{Z< 1 } =.8413 (c) P{ 2 <X< 7 }=P { 2 − 3 4 < X− 3 4 < 7 − 3 4 } = (1)− ...
5.5Normal Random Variables 173 whereris a constant. Ifr=3, andVcan be assumed (to a very good approximation) to be a normal rand ...
174 Chapter 5: Special Random Variables moment generating functions and distributions, we can conclude that ∑n i= 1 Xiis normal ...
5.6Exponential Random Variables 175 1 – a a 0 za FIGURE 5.9 P{Z>zα}=α. The value ofzαcan, for anyα, be obtained from Table A1 ...
176 Chapter 5: Special Random Variables The exponential distribution often arises, in practice, as being the distribution of the ...
5.6Exponential Random Variables 177 function for at least an additional times. Since this will be the case if the total function ...
178 Chapter 5: Special Random Variables However, if the lifetime distributionFis not exponential, then the relevant probability ...
5.6Exponential Random Variables 179 = ∏n i= 1 e−λix =e− ∑n i= 1 λix EXAMPLE 5.6c A series system is one that needs all of its ...
180 Chapter 5: Special Random Variables 0 t n 2 t n 3 t n t = nt n t (n – 1)n FIGURE 5.10 and (e) state that in a small interval ...
5.6Exponential Random Variables 181 with the approximation becoming exact as the number of subintervals,n, goes to∞. However, th ...
182 Chapter 5: Special Random Variables *5.7The Gamma Distribution A random variable is said to have a gamma distribution with p ...
*5.7The Gamma Distribution 183 we see that (n)=(n−1)! The function (α) is called thegammafunction. It should be noted that whenα ...
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