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P1.T2.20.6. Common univariate random variables (normal, lognormal, chi-squared)

Nicole Seaman

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Distinguish the key properties and identify the common occurrences of the following distributions: normal distribution, lognormal distribution, chi-squared distribution ...

Questions:

20.6.1. A hedge fund's big data algorithm can predict the market's direction on five out of eight days (62.5%). Each day's prediction is either a success (e.g., market goes up and algo predicts up) or a failure (e.g., market goes down but algo predicts up). If we dubiously assume the predictions are independent, the binomial distribution fits a series of daily predictions over, say, a week or a month. Over two months, the probability of each day's predication being successful, p, equals 5/8 or 62.5% and the number of days, n, equals 60. We observe that n*p = 60*62.5% = 37.5 and n*(1-p) = 60*37.5% = 22.5, and both of these values (i.e., 37.5 and 22.5) are greater than 10; this satisfies a conventional test that says we can use the normal to approximate the binomial. For example, if p were only 1.0%, then n*p = 6, but 6 is less than 10, and such a binomial is deemed to be too skewed to be approximated by the normal distribution. But ours passes the test so we will approximate with the normal distribution. If we do rely on the normal distribution to approximate this binomial where p = 5/8 and n = 60, what is the probability that the algo makes a correct prediction on only half the days or worse; i.e., where X is the number of successful predictions and we approximate with the normal distribution, what is the Pr(X <= 30)?

a. 2.2750%
b. 8.3500%
c. 11.090%
d. 14.6667%


20.6.2. In human medicine, sometimes the incubation period can be characterized by a lognormal distribution. Let's assume that the incubation period of COVID-19, as measured in days, follows a lognormal distribution and somehow we do know that the mu parameter, µ = 1.10. We don't know the sigma parameter, σ, but we can estimate it from a sample where we observe that the mean (aka, expected value of the) incubation period is 7.0 days. Hint: the mean of a lognormal distribution is given by exp(µ + σ^2/2); see https://en.wikipedia.org/wiki/Log-normal_distribution.

What is the probability that the incubation period will exceed 14.0 days, which is twice as long as the expected duration of one week?

a. Less than 0.0010%
b. 0.982%
c. 11.82%
d. 25.50%


20.6.3. Fourteen (n = 14) experts were asked to forecast gross domestic product (GDP) growth in the year after the post-virus economy is re-opened. Assume the distribution of the forecasted growth rates is normal. The probability is 5.0% that the sample variance is X percent greater than the population variance. We want the value of X. As a hint, recall that we test the sample variance (when the population is normal) with the chi-squared distribution and specifically the quantity df*S^2/σ^2 has a chi-squared distribution with (n-1) degrees of freedom where S^2/σ^2 is the ratio of the sample variance to the population variance. We are looking for Pr[(S^2/σ^2) > (1+X) ] = 5.0%. What is the value of X?

a. 44.8%
b. 50.0%
c. 63.7%
d. 72.0%

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