EX : 3.1 Suppose that X, the reaction time to a stimulus, has a uniform distribution on the interval from 0 to an unknown upper limit 0 (so the density function of X is rectangle in shape with height 1/0 for 0 < x < 0). An investigator wants to estimate 0 on the basis of a random sample X1, ..., X, of reaction times. A very natural estimate is the sample maximum ôs = max(X1,..., Xn). If n = 5, and rı = 4.2, r2 = 1.7, 13 = 2.4, x4 = 3.9, 15 = 1.3, the point estimate of 0 is Ô, = max(4.2, 1.7, 2.4, 3.9, 1.3) = 4.2. Is the estimator ô, an unbiased estimator of 0? Show that; n+1, is an unbiased estimator of 0 using the knowledge of order statistics.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
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