Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 203and 11.2cm, what proportion of the manufactured parts will be rejected onaverage?Answer: If X is used to denote the part length in centimeters, it is reasonableto assume that it is distributed according to N(11, 0.04). Thus, on average, theproportion of acceptable parts is P10:6 < X 11:2). From Equation (7.25),and using Table A.3, we have10:6 11P…10:6 < X 11:2† ˆP < U 0:211:2 110:2ˆ P… 2 < U 1† ˆF U …1† ‰1 F U …2†Šˆ 0:8413 …1 0:9772† ˆ0:8185:The desired answer is then 1 0:8185, which gives 0.1815.The use of the normal distribution in Example 7.3 raises an immediateconcern. Normal random variables assume values in positive and negativeranges, whereas the length of a machine part as well as many other physicalquantities cannot take negative values. However, from a modeling point ofview, it is a commonly accepted practice that normal random variables are validrepresentations for nonnegative quantities in as much as probability P(X < 0)is sufficiently small. In Example 7.3, for example, this probability isP…X < 0† ˆP U < 11 ˆ P…U < 55† 00:2Ex ample 7. 4. Let us compute Pm k