A manufactured item is classified as good, a “second,” or defective with probabilities 6/10, 3/10, and 1/10, respectively. Fifteen such items are selected at random from the production line. Let X denote the number of good items, Y the number of seconds, and 15 − X − Y the number of defective items. (a) Give the joint pmf of X and Y, f (x, y). (b) Sketch the set of integers (x, y) for which f (x, y) > 0. From the shape of this region, can X and Y be independent?Why or why not? (c) Find P(X = 10,Y = 4).
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!
A manufactured item is classified as good, a “second,” or defective with
from the production line. Let X denote the number of
good items, Y the number of seconds, and 15 − X − Y
the number of defective items.
(a) Give the joint pmf of X and Y, f (x, y).
(b) Sketch the set of integers (x, y) for which f (x, y) > 0.
From the shape of this region, can X and Y be
independent?Why or why not?
(c) Find P(X = 10,Y = 4).
(d) Give the marginal pmf of X.
(e) Find P(X ≤ 11).
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