Download e-book for iPad: An Intermediate Course in Probability (Springer Texts in by Allan Gut

By Allan Gut

ISBN-10: 1441901612

ISBN-13: 9781441901613

This is often the one e-book that offers a rigorous and accomplished remedy with plenty of examples, workouts, feedback in this specific point among the normal first undergraduate path and the 1st graduate direction in keeping with degree concept. there's no competitor to this ebook. The booklet can be utilized in study rooms in addition to for self-study.

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Example text

Find the expectation of the remaining piece after it has been broken off n times. 3. That the result E Y = 1/4 is reasonable can intuitively be seen from the fact that X on average equals 1/2 and that Y on average equals half the value of X, that is 1/2 of 1/2. 1 consists, in fact, of a stringent version of this kind of argument. 2. Let X and Y be random variables and g be a function. We have (a) E g(X)Y | X = g(X) · E(Y | X), and (b) E(Y | X) = E Y if X and Y are independent. 4. 2. 4. 1(a)). 2(a) should hold.

Let the joint density function of X and Y be given by f (x, y) = c · x3 y, for x, y > 0, 0, otherwise. x2 + y 2 ≤ 1, Compute c, the marginal densities, and the conditional expectations E(Y | X = x) and E(X | Y = y). 23. The joint density function of X and Y is given by f (x, y) = c · xy, for x, y > 0, 0, otherwise. 4x2 + y 2 ≤ 1, Compute c, the marginal densities, and the conditional expectations E(Y | X = x) and E(X | Y = y). 24. Let X and Y have joint density   c , when 1 < y < x, f (x, y) = x3 y 0, otherwise.

However, the object of interest is a one-dimensional quantity; the problem is formulated in terms of the joint distribution of X and Y , and we are interested in the distribution of X (the density fX (x)). Before we solve this problem we shall study the discrete case, which, in some respects, is easier to handle. Thus, suppose that (X, Y ) is a given two-dimensional random variable whose joint probability function is pX,Y (x, y) and that we are interested in finding pX (x). We have {X = x, Y = y}) pX (x) = P (X = x) = P ( y P (X = x, Y = y) = = y pX,Y (x, y).

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An Intermediate Course in Probability (Springer Texts in Statistics) by Allan Gut


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