- For a given continuous random variable, the discrete value of Z is always undefined. Why? 2) For a continuous uniform distribution from 0 to 12, what is the mean? 3) For a binomial distribution B(10,.75), what is the mean? What is the variance? 4) You are evaluating the probability that Team A will win 14 out of it’s 20 games this season. You correctly used the formula [p^x] * [(1-p)^(x-n)] * nCr to determine the probability of winning 14 games. What have you forgotten to do and why is your answer grossly understated? 5) True or False, the normal distribution is complete defined by it’s mean of 0, standard deviation of 0, skewness of 0 and kurtosis of 3? 6) You have a sample with 100 observations? What is the strongest statement you can make regarding the number of observations that fall within 2 standard deviations of the mean? 7) A normally distributed samples have approximately 90% of observation within +/- 1.65 std of the mean, 95% within 1.96 of std of the mean, and 99% within 2.68 std of the mean. If this sample were non-normally distributed, the variance was known, and you had a large number of observations… would your confidence interval be larger or smaller at these given std deviations? 8) When would you use the lognormal distribution? Bonus: What happens to the mean value of the lognormal distribution as variance increases?
Obviously not as popular a section, but here goes; ----------------------------------------------------------------------- 1) For a given continuous random variable, the discrete value of Z is always undefined. Why? The value at an given point is zero because the distribution is infinitely divided. You need to defined the value of a range. 2) For a continuous uniform distribution from 0 to 12, what is the mean? The mean is equal to (b-a)/2 = 6 3) For a binomial distribution B(10,.75), what is the mean? What is the variance? The mean is n * p —> 10 * .75 -----> 7.5 The variance is np(1-p) —> 1.875 4) You are evaluating the probability that Team A will win 14 out of it’s 20 games this season. You correctly used the formula [p^x] * [(1-p)^(x-n)] * nCr to determine the probability of winning 14 games. What have you forgotten to do and why is your answer grossly understated? You must remember to add up the individual probability values for winning exactly 1 game, plus winning exactly 2 games… up to 14 to get the correct cumulative distribution. 5) True or False, the normal distribution is complete defined by it’s mean of 0, standard deviation of 0, skewness of 0 and kurtosis of 3? One part that is false is that it should read variance of 1, not standard deviation of 0. 6) You have a sample with 100 observations? What is the strongest statement you can make regarding the number of observations that fall within 2 standard deviations of the mean? Unless it is specified that it is a normal distribution, you would need to use chebyshev’s inequality. In this case 1-(1/2^2) = 75%. So 75 observations. 7) A normally distributed samples have approximately 90% of observation within +/- 1.65 std of the mean, 95% within 1.96 of std of the mean, and 99% within 2.68 std of the mean. If this sample were non-normally distributed, the variance was known, and you had a large number of observations… would your confidence interval be larger or smaller at these given std deviations? If you had a non-normal distribution with a known variance and a large number of observations, you could use the Z-alt of T-tests. Z will give you a tighter distribution than the t-test. However, both will be wider than a normal distribution. 8) When would you use the lognormal distribution? Practical application is for stock prices – because you cannot have values less than zero. Bonus: What happens to the mean value of the lognormal distribution as variance increases? The mean will actually increase when the variance increases. This is because the distribution is bound at zero.