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Math 140 - Cooley Business Calculus OCC Section 5.4 – Probability Definition Let x be a continuous random variable. A function f is said to be a probability density function for x if: 1. For all x in the domain of f, we have 0 f ( x ) . 2. The area under the graph of f is 1. 3. For any subinterval [c, d ] in the domain of f, the probability that x will be in that subinterval is given by d P([c, d ]) f ( x )dx . c Definition A continuous random variable x is said to be uniformly distributed over an interval [a, b] if it has a probability density function f given by 1 , a xb. f ( x) ba Definition A continuous random variable is exponentially distributed if it has a probability density function of the form f ( x) ke kx , over the interval [0, ) . Exercises: Verify Property 2 of the definition of a probability density function over the given interval. 1) f ( x) 2 x , 2) 1 f ( x) x 2 , 3 [0,1] [2,1] -1- Math 140 - Cooley Business Calculus OCC Section 5.4 – Probability Exercises: Find k such that each function is a probability density function over the given interval. Then write the probability density function. 3) f ( x ) kx , [1,4] 4) f ( x) ke x , [0,3] 5) A number is selected at random from the interval [5,29] . The probability density function for x is given by 1 , for 5 x 29 . f ( x) 24 Find the probability that a number selected is in the subinterval [14,29] . 6) A telephone company determines that the duration t, in minutes, of a phone call is an exponentially distributed random variable with a probability density function f (t ) 2e2t , for 0 x . a) Find the probability that a phone call will last no more than 2 minutes. b) Find the probability that a phone call will last more than 5 minutes. -2-