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Chapter 5: Q. 5.20 (page 218)
For any real number , define by
Let be a constant.
(a) Show that
when is a standard normal random variable.
(b) Find when is normal with mean and variance .
a. Using the concept of a random variable's expectation it is proved.
b. The value of is.
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Verify thatwhenis a gamma random variable with parameters and
An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with and , whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters . A point is randomly chosen on the image and has a reading of . If the fraction of the image that is black is , for what value of would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?
An image is partitioned into two regions, one white and the other black. A reading taken from a randomly chosen point in the white section will be normally distributed with and, whereas one taken from a randomly chosen point in the black region will have a normally distributed reading with parameters. A point is randomly chosen on the image and has a reading of. If the fraction of the image that is black is, for what value of would the probability of making an error be the same, regardless of whether one concluded that the point was in the black region or in the white region?
Let X and Y be independent random variables that are both equally likely to be either 1, 2, . . . ,(10)N, where N is very large. Let D denote the greatest common divisor of X and Y, and let Q k = P{D = k}.
(a) Give a heuristic argument that Q k = 1 k2 Q1. Hint: Note that in order for D to equal k, k must divide both X and Y and also X/k, and Y/k must be relatively prime. (That is, X/k, and Y/k must have a greatest common divisor equal to 1.) (b) Use part (a) to show that Q1 = P{X and Y are relatively prime} = 1 q k=1 1/k2 It is a well-known identity that !q 1 1/k2 = 蟺2/6, so Q1 = 6/蟺2. (In number theory, this is known as the Legendre theorem.) (c) Now argue that Q1 = "q i=1 P2 i 鈭� 1 P2 i where Pi is the smallest prime greater than 1. Hint: X and Y will be relatively prime if they have no common prime factors. Hence, from part (b), we see that Problem 11 of Chapter 4 is that X and Y are relatively prime if XY has no multiple prime factors.)
The speed of a molecule in a uniform gas at equilibrium is a random variable whose probability density function is given by
whereanddenote, respectively,
Boltzmann鈥檚 constant, the absolute temperature of the gas,
and the mass of the molecule. Evaluate in terms of.
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