5- Suppose X, and X2 are i.id observations from the pdf fa,ß (x) = axa-1e-x" ,x>0, a > 0 Show that logx, is an ancillary statistics. logX2
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- Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?Suppose the time it takes Alex to do this exam is exponentially distributed with parameter 3 per hour, and the time it takes Ben to do the exam is exponentially distributed with parameter 2per hour. Assume that these two times are independent.(a) What is the probability that Alex finishes before Ben?(b) What is the expected time in minutes until the first one finishes this exam?(c) What is the probability that neither Alex nor Ben finishes the exam within 3 hours?
- A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.Suppose that X is a continuous unknown all of whose values are between -3 and 3 and whose PDF, denoted f , is given by f ( x ) = c ( 9 − x^2 ) , − 3 ≤ x ≤ 3 , and where c is a positive normalizing constant. What is the variance of X?Suppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?
- Suppose X1, . . . , Xn ∼ Exponential(λ) is a set of n observations drawn independently from an Exponential distribution. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?Let X1 and X2 be IID exponential with parameter > 0. Determine the distribution ofY = X1=(X1 + X2).
- Consider a cohort of patients receiving an experimental two-year treatment for a serious disease. Suppose that only 20% of the patients survive to the end of the second year. Assume CFM (constant force of mortality) within each year of treatment. Find the probability that a patient who survives the first four months dies by the end of the 20th month. Round to 4 decimal places.The lifetime of a certain type of TV remote control is given by Y . Suppose Y has approximately exponential distribution with mean 8 years. a) Find the probability that a remote control of this type will last more than 15 years. b) Find the probability that of eight such remote controls at least one will last more than 15 years. c) What should the warranty period for these remote controls be if the manufacturer wants 85% of the remote controls to last beyond the warranty period? d) What is the moment generating function of Y .The positive random variable X is said to be a log-normal random variable with parameters μ and σ2 if log(X) is a normal random variable with mean μ and variance σ2. Use the normal moment generating function to find the mean and variance of a lognormal random variable. Hint: Let Y = log(X ) and find E[X ].
