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MAT 1235 Calculus II Section 8.5 Probability http://myhome.spu.edu/lauw HW WebAssign 8.5 (6 problems, 65 min.) Quiz: 8.2, 8.5 Preview Provide a 30-minute snapshot of probability theory and its relationship with integration. Preview Provide a 30-minute snapshot of probability theory and its relationship with integration. Engineering: MAT2200 (3) Math major/minor: MAT 3360 (5) Random Variables Variables related to random behaviors Example 1 Y=outcome of rolling a die = X=lifetime of a Dell computer = Q: What is a fundamental difference between X and Y? Continuous Random Variables Take range over an interval of real numbers. Probability… of an event = the chance that the event will happen Example 2 P(Y=1)=1/6 The chance of getting “1” is ___________ P(3≤X≤4) The chance that the Dell computer breaks down____________________ Probability… …of an event = the chance that the event will happen …is always between 0 and 1. Example 3 P(Y=7)= P(0 ≤ X<)= Probability Density Function Continuous random variable X The pdf f(x) of X is defined as b P a X b f x dx a The prob. info is “encoded” into the pdf Probability Density Function Properties: 1. f x 0 for all x 2. f x dx 1 Example 4 1 2 4 x 12 x if 0 x f x 2 0 Otherwise (a) Show that f(x) is a pdf of some random variable X. Example 4 1 2 4 x 12 x if 0 x f x 2 0 Otherwise (b) Let X be the lifetime of a type of battery (in years). Find the probability that a randomly selected sample battery will last more than ¼ year. 13 0.8125 16 Average Value of a pdf xf x dx Also called 1. Mean of the pdf f(x) 2. Expected value X Example 4 1 2 4 x 12 x if 0 x f x 2 0 Otherwise (c) Let X be the lifetime of a type of battery (in years). Find the average lifetime of such type of batteries. xf x dx 17 0.354 years 48 Exponential Distribution 0 f t ct ce if t 0 if t 0 Used to model waiting times, equipment failure times. It have a parameter c. The average value is 1/c. So, c = Example 5 The customer service at AT&T has an average waiting time of 2 minutes. Assume we can use the exponential distribution to model the waiting time. Find the probability that customer will be served within 5 minutes. Example 5 Let T be the waiting time of a customer. 0 f t ct ce if t 0 if t 0 0.92 Remarks If a random variable is not given, be sure to define it.