Suppose that Y1, ...,Y2m+1 are the order statistics from a random sample from an Exponential(1) random variable. a. Find

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Suppose That Y1 Y2m 1 Are The Order Statistics From A Random Sample From An Exponential 1 Random Variable A Find 1 (85.15 KiB) Viewed 86 times

Suppose that Y1, ...,Y2m+1 are the order statistics from a random sample from an Exponential(1) random variable. a. Find the pdf of the smallest order statistic, Y . b. What is the probability that Y, exceeds 1? Your answer will be a function of the sample size. C. Find the pdf of the sample median (Hint: Because our sample size is 2m + 1, the median will be Ym+1. Feel free to use the proposition on page 277 directly) d. Suppose now we have a sample of size 101, so that m = 50. Integration (using software) will show that E[Y51] = 0.6981. An exact value can be found but it is a fraction with 43 digits in the numerator and denominator, so it is omitted here. What property of the Exponential(1) distribution makes it so the expected value of the median is not equal to the mean of the Exponential(1) distribution (which is 1).