Ps4

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School
Arizona State University **We aren't endorsed by this school
Course
STP 421
Subject
Statistics
Date
Aug 31, 2023
Pages
3
Uploaded by DeanKnowledge27492 on coursehero.com
STP 421 - Spring 2023 Problem Set 4 Due: April 12 2023 Instructions: Show your work and attempt every problem! Partial credit will be given if you can show some progress towards a solution. If you can't solve a problem in full generality, try to solve some special cases. You may work in groups and consult outside resources (textbooks, web sites, etc.). How- ever, you should acknowledge any assistance received from other people or outside re- sources. If working in a group, please list the names of your group members. Submitted solutions should either be typed, preferably using LaTeX, or neatly handwrit- ten. Solutions should be submitted as PDF documents through Canvas. If submitting scanned images, please assemble all pages into a single PDF document prior to uploading to Canvas. 1
1.) Suppose that X is a continuous random variable with density p X ( x ) = ( Cx (1 - x ) if x [0 , 1] 0 if x < 0 or x > 1 . (a) Find C so that p X is a probability density function. (b) Find the cumulative distribution of X . (c) Calculate the probability that X (0 . 1 , 0 . 9). (d) Calculate the mean and the variance of X . 2.) Suppose that X is a continuous random variable with cumulative distribution function F X ( x ) = 1 π arctan( x ) + 1 2 . (a) Find the probability density function of X . (b) Calculate the probability that X 1. (c) Show that the moment generating function m X ( t ) = for any t 6 = 0. 3.) Let V = 1 - U , where U is a standard uniform random variable. Find the cumulative distribution function and the probability density function of V and use your results to conclude that V is also a standard uniform random variable. 4.) Suppose that X is exponentially distributed with parameter 1 and let Y = γX , where γ > 0 is a positive constant. Find the cumulative distribution function and the density of Y and use this to identify the distribution of Y . 5.) Assuming that the time that it takes a radioactive nucleus to decay is exponentially dis- tributed, find the relationship between the rate of decay λ and the half-life t 1 / 2 , which is the time at which the probability of decay is equal to 1 / 2. Given that the half-life of carbon-14 is 5730 years, approximately what proportion of a sample of carbon-14 will remain after 44,000 years? 6.) Suppose that X 1 , · · · , X n are independent exponential random variables with parameters λ 1 , · · · , λ n . Find the distribution of Y = min { X 1 , · · · , X n } . Hint: Calculate the probability P ( Y > t ). 2
7.) Suppose that X is a random variable with values in [0 , 1] and that E [ X ] = p . Show that V ar ( X ) p (1 - p ). 8.) Suppose that the value of the random variable X is determined in two steps. First, a fair coin is tossed. If the coin lands on heads, then the value of X is chosen uniformly from the interval [0 , 2]. Otherwise, if the coin lands on tails, then the value of X is chosen uniformly from the interval [ - 1 , 0]. (a) Find the probability density function of X . (b) Calculate the probability of the event | X | > 1 / 2. (c) Calculate the expected value of X . (d) Calculate the variance of X . 9.) Suppose that the distribution of body temperatures (measured in F ) in healthy awake adults is approximately normal with mean 98 . 6 and variance 0 . 49. (a) Write down the density of this distribution. (b) Express the probability that an individual's temperature is between 97 . 9 F and 100 F in terms of the cumulative distribution function of the standard normal distribution. (c) What is the distribution of body temperature measured in C ? (Recall that C = ( F - 32) · 5 / 9 . ) 10.) Morphological traits such as wing and tail length can be used to determine the sex of some adults in bird species lacking sexually dimorphic plumages. For example, a study of adult Spotted Owls living in California found that the mean and the standard deviation of wing length (measured in mm) are approximately 329 and 6 in females and 320 and 6 in males (Blakesley et al., 1990, J. Field Ornithology). Assuming that male and female owls are equally abundant and that wing length is normally distributed within each sex, (a) find the probability density of the wing length X of an individual sampled at random from this population; (b) calculate the mean and the variance of X ; (c) calculate the probability that a randomly sampled individual is male given that the wing length is 320 mm. 3
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