(Proof of the independence of X and S2 for n = 2) IfX1 and X2 are independent random variables having thestandard normal distribution, show that(a) the joint density of X1 and X is given by f(x1, x) = 1π · e−x−2e−(x1−x)2for −q < x1 < q and −q < x < q;(b) the joint density of U = |X1 − X| and X is given by g(u, x) = 2π · e−(x2+u2) for u > 0 and −q < x < q, since f(x1, x) is symmetricalabout x for fixed x;(c) S2 = 2(X1 − X)2 = 2U2;(d) the joint density of X and S2 is given byh(s2, x) = 1√π e−x2· 1√2π(s2)− 12 e− 12 s2for s2 > 0 and −q < x < q, demonstrating that X and S2are independent.
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!
X1 and X2 are independent random variables having the
standard
(a) the joint density of X1 and X is given by
π · e−x−2
e−(x1−x)2
for −q < x1 < q and −q < x < q;
(b) the joint density of U = |X1 − X| and X is given by
π · e−(x2+u2)
about x for fixed x;
(c) S2 = 2(X1 − X)2 = 2U2;
(d) the joint density of X and S2 is given by
h(s
2, x) = 1
√π e−x2
· 1
√
2π
(s
2)
− 1
2 e− 1
2 s2
for s2 > 0 and −q < x < q, demonstrating that X and S2
are independent.
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