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Unformatted text preview: Stat 400 In Class Worksheet 6 T.A. Emily King March 12, 2008 1. (Modified from book p.143 # 13): Let X be the continuous random variable for X = the headway beyween two randomly selected consecutive cars (in seconds) Suppose that in a certain traffic environment, the distribution of time headway has the form k x4 f (x) = x>1 x1 0 (a) Determine the value of k for which f (x) is a legitimate pdf. (b) Obtain the cdf. (c) Use the cdf to determine the probability that headway exceeds 2 sec and also the probability that headway is between 2 and 3 sec. (d) Obtain the mean value of headway and the standard deviation of headway. (e) What is the probability that headway is within 1 standard deviation of the mean value? (f) What is the median value of the distribution? 2. (Book p.19 #19): Let X be a continuous rv with cdf 0 x0 1 + ln
4 x F (x) = x 4 0<x4 x>4 1 (This is a cdf which has been used as a model for a certain hyrdologic variable.) What is (a) P (X 1)? (b) P (1 X 3)? (c) The pdf of X? 3. (Tricky) Let X be a continuous random variable with values between A = 1 and B = b and with density function f (x) = ln(x). What value must b be in order for f to be a probability density function? 4. (Math 220/221 book p 607 Ex 5 & p 613 Ex 3) Let X be the random variable associated with the experiment that consists of selecting a point at random from a circle of radius 1 and observing its observing its distance from the center. (a) Find the probability density function f (x) and the cumulative density function F (x) of X. (b) Find the expected value, variance and standard deviation. 5. Find the values of k that make 3kx(x + 4k) a probability density function for 0 x 1. ...
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