STAT 400 Lecture AL1 1 Spring 2015 Dalpiaz Practice Problems 3 Suppose a random variable X has the following probability density function f x sin x 0 x 2 Find E X a Find P X c Find the median of the probability distribution of X 2 Suppose a random variable X has the following probability density function 4 zero otherwise f x x e x b 0 x 1 1 2 zero otherwise Find E X a Find P X c Find the moment generating function of X M X t 3 Let X have the probability density function b f x 2 x for 0 x 1 f x 2 for 1 x 2 f x 0 otherwise 3 3 a Verify that f x is a probability density function b Find P X 0 5 d Find P X 1 25 e Find the cumulative distribution function F x P X x f Find the median of the probability distribution of X g Compute the expected value of X E X X c Find P X 1 5 4 The heights of adult males in Neverland are normally distributed with mean of 69 inches and standard deviation of 5 inches a What proportion of adult males in Neverland are taller than 6 feet 72 inches b What proportion of adult males are between 5 and 6 feet tall c How tall must a male be to be among the tallest 10 of the population d How tall must a male be to be among the shortest third of the population 5 The lifetime of a certain type of television tube is normally distributed with mean 3 8 years and standard deviation of 1 2 years a Suppose that the tube is guaranteed for two years What proportion of TVs will require a new tube before the guarantee expires b If the company wishes to set the warranty period so that only 4 of the tubes would need replacement while under warranty how long a warranty must be set c What proportion of TVs will last for over 5 years 6 The lifetimes of a certain type of light bulbs follow a normal distribution If 90 of the bulbs have lives exceeding 2000 hours and 3 have lives exceeding 6000 hours what are the mean and standard deviation of the lifetimes of this particular type of light bulbs 7 Suppose the average daily temperature in degrees Fahrenheit in July in Anytown is a random variable T with mean T 86 and standard deviation T 9 What is the probability that the daily temperature in July in Anytown is above 33 degrees Celsius Assume that T is a normal random variable Celsius Fahrenheit F 9 C 32 5 Fahrenheit Celsius C 5 F 32 9 8 If the moment generating function of X is M t exp 166 t 200 t 2 find a The mean of X b The variance of X c P 170 X 200 d P 148 X 172 From the textbook 9 3 1 8 a 3 3 2 a 3 3 4 a 3 2 2 a 10 3 1 8 b 3 3 2 b 3 3 4 b 3 2 2 b 11 3 1 8 c 3 3 2 c 3 3 4 c 3 2 2 c 12 3 1 10 3 3 8 3 2 8 13 3 1 4 3 4 4 3 3 4 14 3 2 2 3 4 8 3 3 8 15 3 3 6 3 6 6 5 2 6 16 3 3 11 3 6 14 5 2 14 Answers 1 Suppose a random variable X has the following probability density function f x sin x a Find P X P X b 4 4 0 x 2 zero otherwise 4 sin x dx cos x 0 4 0 1 4 2 0 292893 2 Find E X E X 2 x sin x dx x cos x sin x 0 c 1 cos 2 1 0 Find the median of the probability distribution of X x F x P X x sin y dy 1 cos x 0 x 0 Median F m P X m cos m 1 2 1 2 m 3 2 2 Suppose a random variable X has the following probability density function f x x e x a Find P X P X b 0 x 1 zero otherwise 1 2 12 1 2 x e x dx x e x e x 0 12 1 0 2 0 175639 Find E X E X 1 x x e x dx x 2 e x 2 x e x 2 e x 0 c e Find the moment generating function of X M X t 1 MX t e t x x e x dx 0 1 x e t 1 x dx 0 1 1 x e t 1 x e t 1 x 2 1 t t 1 1 1 1 e t 1 e t 1 2 t 1 t 1 t 1 2 t e t 1 1 t 1 2 1 MX 1 0 x dx t 1 1 2 1 0 1 0 e 2 3 a x 0 total area under the graph of x is 1 b P X 0 5 1 12 c P X 1 5 2 3 d P X 1 25 0 e F x 0 for x 0 1 2 for 0 x 1 x 3 2 1 for 1 x 2 F x x 3 3 F x F x 1 for x 2 f median 1 25 g E X 11 9 4 The heights of adult males in Neverland are normally distributed with mean of 69 inches and standard deviation of 5 inches a What proportion of adult males in Neverland are taller than 6 feet 72 inches 72 69 P X 72 P Z 5 P Z 0 60 area to the right of 0 60 1 0 60 1 0 7257 0 2743 b What proportion of adult males are between 5 and 6 feet tall 72 69 60 69 P 60 X 72 P Z 5 5 P 1 80 Z 0 60 area between 1 80 and 0 60 0 60 1 80 0 7257 0 0359 0 6898 c How tall must a male be to be among the tallest 10 of the population Need x such that P X x 0 10 Need z such that P Z z 0 10 z 1 0 10 0 90 z 1 28 x x 69 z 1 28 5 x 69 5 1 28 75 4 inches A male must to be over 75 4 inches tall d How tall must a male be to be among the shortest third of the population Need x such that P X x 0 3333 Need z such that P Z z 0 3333 z 0 3333 z 0 43 x x 69 z 0 43 5 x 69 5 0 43 66 85 inches A male must to be under 66 85 inches tall 5 3 8 …
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