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A man invests a total of N dollars in a group of n securities, whose rates of return (interest rates) are independent random variables X1, X2, .........., Xn, respectively, with means i1, i2, ... , in and variances σl2, σ22, ... , an2, respectively. If the man invests Nj dollars in the jth security, then his return in dollars on this particular portfolio is a random variable R given by R = N1X1 + N2X2 + ... + NnXn. Let the standard deviation σ[R] of R be used as a measure of the risk involved in selecting a given portfolio of securities. In particular, let us consider the problem of distributing investments of 5500 dollars between two securities, one of which has a rate of return Xv with mean 6 % and standard deviation 1 %, whereas the other has a rate of return X2 with mean 15% and standard deviation 10%. (i) If it is desired to hold the risk to a minimum, what amounts N1 and N2 should be invested in the respective securities? What is the mean and variance of the return from this portfolio? (ii) What is the amount of risk that must be taken in order to achieve a portfolio whose mean return is equal to 400 dollars? (iii) By means of Chebyshev's inequality, find an interval, symmetric about 400 dollars, that, with probability greater than 75 %, will contain the return R from the portfolio with a mean return E[R] = 400 dollars. Would you be justified in assuming that the return R is approximately normally distributed?
Is it likely that a standard normal random variable will have a value less than -4? Explain.- Find a value such that the probability that the standard normal random variable will be above it is 0.85.
The candy company claims that 10% of the M&M's it produces are green. Suppose that the candies are packaged at random in small bags containing about 50 M&M's. A class of elementary school students learning about percents opens several bags, counts th..
David Gallano, a wine merchant, has collected opinions on grape wine quality from a sample of his customers. These customers tasted wine made from grapes grown in three regions of the country.
Monitoring the ecological health of the Everglades-the bottom temperatures are recorded at the Garfield Bight station and the mean of 19.7 degrees C is obtained for 62 temperatures on 62 different days.
According to Harper's Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 16 federal inmates is selected:
the probability that it gets by the first inspector is 0.1. Of those that get past the first inspector, the second inspector will miss 5 out of 10. What fraction of the defective items gets by both inspectors?
Is there any evidence the true mean number of hours worked by white-collar employees is greater than 40? Use a = 0.01. What assumption(s) did you make in order.
It is known that the standard deviation of the population of checkout times is one minute. With a .95 probability, the sample mean will provide a margin of error of?
The book Red State, Blue State, Rich State, Poor State by Andrew Gelman [13] discusses the following election phenomenon: within any U.S. state, a wealthy voter
Tell whether each of these features characterizes a paired t confidence interval, a two-sample t confidence interval, or both, or neither:
question as a sample size approaches infinity how does the students t distribution compare to the normal z
as we have noted in previous chapters even a very small effect can be significant if the if the sample is large enough.
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