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An Introduction to Mathematical Statistics and Its Applications (6th Edition)
- 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?arrow_forwardX, Y follow the exponential distribution with rate parameter lambda = 5.0 P[X >= 5] ? Write P[X+Y <= 14] as an integralarrow_forwardSuppose 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?arrow_forward
- Let X be a (continuous) uniform random variable on the interval [0,1] and Y be an exponential random variable with parameter lambda. Let X and Y be independent. What is the PDF of Z = X + Y.arrow_forwardLet Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size nfrom a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that thekth order statistic Yk has a beta pdf with parameters α = k and β = n − k + 1.arrow_forwardIn certain experiments, the error made in determiningthe density of a substance is a random variable having auniform density with α = −0.015 and β = 0.015. Find theprobabilities that such an error will(a) be between −0.002 and 0.003;(b) exceed 0.005 in absolute value.arrow_forward
- Suppose X and Y are independent, exponentially distributed random variables with rate parameter λ, λ > 0. Find the joint PDF of U and V , where U = X + Y, V = X/Y.arrow_forwardA college professor never finishes his lecture before the end of the hour and always finishes his lectures within 2 min after the hour. Let X = the time that elapses between the end of the hour and the end of the lecture and suppose the pdf of X is f(x)= {kx2O < x < 2} 0 otherwise a. Find the value of k and draw the corresponding density curve.[ Hint: Total area under the graph of f (x) is 1.] b. What is the probability that the lecture ends within 1min of the end of the hour? c. What is the probability that the lecture continues beyond the hour for between 60 and 90 sec? d. What is the probability that the lecture continues for at least 90 sec beyond the end of the hour?arrow_forwardWhile taking a walk along the road where you live, you accidentally drop your glove, but you don't know where. The probability density p(x) for having dropped the glove x kilometers from home (along the road) is p(x)=7e−7x for x≥0 At what distance y from home is the probability that you dropped it within y km of home equal to 0.95?arrow_forward
- The molecular weight of a particular polymer should fall between 750 and 1,035. Thirty samples of this material were analyzed with the results X-bar = 902 and s = 90. Assume that molecular weight is normally distributed.arrow_forwardThe random variable Z has a uniform distribution in the unit interval and the X is uniform on [z, z + 1]:• Draw the region of (X, Z) where their density function is defined.• What is the probability of 0.5 < X?• What is the expected value of X?• What is the probability of 0.5 < Z?• How you can interprept their correlation QUANTIATIVELY?arrow_forwardThe area under the standard normal curve from zero to z is 0.4505 and z is positive. The value of z isarrow_forward
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