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STT 231 – 01 PRACTICE SHEET SET 5 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DENSITY FUNCTION, p.d.f. Example 1: Suppose that the error in the reaction temperature, in oC, for a controlled laboratory experiment is a continuous random variable X having the function x2 , 1 x 2 f ( x) 3 0, elsewhere. (a) Show that f(x) is a density function. (b) Find P(0 < X < 1). Example 2 The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function 2( x 2) , 0 x 1 f ( x) 5 0, elsewhere. (a) Show that P( 0 < X < 1) = 1 (b) Find the probability that more than 1 1 but fewer than of the people contacted will 4 2 respond to this type of solicitation. SOME CONTINUOUS DISTRIBUTION FUNCTIONS 1. UNIFORM DISTRIBUTION FUNCTION A continuous random variable, r.v. X is said to have a uniform distribution on the interval [a,b] if the probability density function, p.d.f. of X is 1 , a xb f ( x) b a 0, otherwise. 1 Example 3: A continuous random variable X that can assume values between x = 1 and x 1 = 3 has a density function given by f ( x) . (a) Show that the area under the curve is 2 equal to 1. (b) Find P( 2 < X < 2.5). (c) Find P ( X 1.6). 2. EXPONENTIAL DISTRIBUTION FUNCTION The continuous random variable X has an exponential distribution, with parameter , if its density function is given by for 0. e x , f ( x) 0, for x 0 otherwise. Example 4: Suppose X is an exponentially distributed random variable with parameter 3 . Find the following probabilities: (a) P( X 2) (b) P( X 1.5) (c) P( X 3) (d ) P( X 0.45) Example 5: Suppose X has an exponential distribution with 2.5. Find the following probabilities: (a) P( X 3) (b) P( X 4) (c) P( X 1.6) (d ) P( X 0.4) 3. NORMAL DISTRIBUTION FUNCTION, N ( , 2 ) Let and 0 be constants. The density function f given by ( x 2) e 2 , f ( x) 2 0, 2 is called the normal density with parameters for x otherwise. and . A special case of the normal density function when 0 and 1 is the standard normal density function denoted by N(0,1). 2 If X is a continuous random variable, its standard normal density function is given by x e 2 , x f ( x) 2 0, otherwise. 2 DEFINITION: If X is a random variable whose density is normal with parameters and , then, X Z is a random variable with a standard normal density. REMARKS: Probably the most important continuous distribution is the normal distribution which is characterized by its “bell-shaped” curve. The mean is the middle value of this symmetrical distribution. When we are finding probabilities for the normal distribution, it is a good idea first to sketch a bell-shaped curve. Next, we shade in the region for which we are finding the area, i.e., the probability. [Areas and probabilities are equal] Then use a standard normal table to read the probabilities. Example 6: Let Z have a standard normal distribution N(0,1). Find the following probabilities: (a) P(0 Z 1.43) (b) P(0 Z ) (c) P( Z 1.61) (d ) P(1.52 Z 1.43) Sketch a bell-shaped curve and shade the area under the curve that equals the probabilities. Example 7: DO EXERCISES 20.1, 20.3 TEXT PAGE 138. REMARK: In statistical applications, we are often interested in right-tail probabilities. We let z be a number such that the probability to the right of z is . That is, P ( Z z ) DIAGRAM 3 (b) P(Z 1.64) Example 8: Find (a) z 0.025 (c) z 0.05 Example 9: Exercises 20.2; Text page 138 MORE PRACTICE PROBLEMS ON DISCRETE AND CONTINUOUS DISTRIBUTION FUNCTIONS 1. Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X: (a) f ( x) c( x 2 4) for x 0,1, 2, 3; [1/30] (b) 2 3 f ( x) c x 3 x for x 0,1, 2. [ 1 ] 10 2. The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function 20,000 , x0 f ( x) ( x 100) 3 0, elsewhere. Find the probability that a bottle of this medicine will have a shelf life of 1 (a) at least 200 days; [ ] 9 (b) anywhere from 180 to 120 days. [0.1020] 3. The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function 0 x 1 x, f ( x) 2 x, 1 x 2 0, elsewhere. Find the probability that over a period of one year, a family runs their vacuum cleaner (a) less than 120 hours; [0.68] 4 (b) between 50 and 100 hours. [0.375] 4. A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by 2(1 x) f ( x) 27 Find (a) P( X 4) [ 16 ] 27 1 [ ] 3 (b) P(3 X 4) 5. Consider the density function k x f ( x) 0, Evaluate k. [ 0 x 1 elsewhere. 3 ] 2 6. Let X denote the amount of time for which a book on two-hour reserve at a college library is checked out by a randomly selected student, and suppose that X has density function x , f ( x) 2 0, Calculate (a) P( X 1) [[0.25] 0 x2 otherwise. (b) P(0.5 X 1.5) [0.50] (c) P(1.5 X ) [0.4375] 7. Suppose the reaction temperature X (in oC) in a certain chemical process has a uniform distribution with A = - 5 and B = 5. Compute: (a) P( X 0) (b) P(2 X 2) (c) P(2 X 3) (d) For k satisfying 5 k k 4 5, compute P(k X k 4) . 5 8. Suppose the distance X between a point target and a shot aimed at the point in a coinoperated target game is a continuous random variable with p.d.f. 3(1 x 2 ) f ( x) 4 0, 1 x 1 otherwise (a) Sketch the graph of f(x). Compute: (b) P( X 0) [0.50] (d ) P( X 0.25 or (c) P(0.5 X 0.5) [0.6875] X 0.25) [0.6328] 9. Let X be a random variable with a standard normal distribution. Find 1 (ii) P(0.53 X 2.03) (iii) P( X 0.73) (iv) P ( X ). 4 10. Let X be normally distributed with mean 8 and standard deviation 4. Find: (i) P(0.81 X 1.13) (i) P(5 X 10). (ii) P (10 X 15), (iii) P ( X 15), (iv) P( X 5) . 11. A fair die is tossed 180 times. Find the probability that the face 6 will appear (i) between 29 and 32 times inclusive, [0.3094] (ii) between 31 and 35 times inclusive. [0.3245]. The answers provided above are when the Binomial Model is used. What if you use the Normal Model? 12. Among 10,000 random digits, find the probability that the digit 3 appears at most 950 times. [0.0475] 13. Suppose the temperature T during June is normally distributed with mean 68o and standard deviation 6o. Find the probability that the temperature is between 70o and 80o. [0.3479] 14. Suppose the heights H of 800 students are normally distributed with mean 66 inches and standard deviation 5 inches. Find the number N of students with heights (i) between 65 and 70 inches, [294] (ii) greater than or equal to 6 feet (72 inches) [92]. 15. Let X be a random variable with a standard normal distribution. Determine the value of t if (i) P(0 X t ) 0.4236 [t = 1.43] (ii) P( X t ) .7967 [t = 0.83] (iii) P(t X 2) 0.1000 [t = 1.16] 6