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Chapter 15: Q23P (page 744)
Do Problem if one person is busy evenings, one is busyevenings, two are each busy one evening, and the rest are free every evening.
The required probability is .
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Let be independent random variables, each with density function , expected value , and variance . Define the sample meanby.Showthat,and . (See Problems and.)
(a) Set up a sample space for the 5 black and 10 white balls in a box discussed above assuming the first ball is not replaced. Suggestions: Number the balls, say 1 to 5 for black and 6 to 15 for white. Then the sample points form an array something like (2.4), but the point 3,3 for example is not allowed. (Why?
What other points are not allowed?) You might find it helpful to write the
numbers for black balls and the numbers for white balls in different colors.
(b) Let A be the event “first ball is white” and B be the event “second ball is
black.” Circle the region of your sample space containing points favorable to
A and mark this region A. Similarly, circle and mark region B. Count the
number of sample points in A and in B; these are and . The region
AB is the region inside both A and B; the number of points in this region is
. Use the numbers you have found to verify (3.2) and (3.1). Also find
and and verify (3.3) numerically.
(c) Use Figure 3.1 and the ideas of part (b) to prove (3.3) in general.
Find the number of ways of puttingparticles in boxes according to the three kinds of statistics.
(a) Find the probability density function for the position x of a particle which is executing simple harmonic motion on along the x axis. (See Chapter , Section , for a discussion of simple harmonic motion.) Hint: The value of x at time t is . Find the velocity ; then the probability of finding the particle in a given is proportional to the time it spends there which is inversely proportional to its speed there. Don’t forget that the total probability of finding the particle somewhere must be .
(b) Sketch the probability density function found in part (a) and also the cumulative distribution function [see equation ].
(c) Find the average and the standard deviation of x in part (a).
A coin is tossed repeatedly; x = number of the toss at which a head first appears.
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