By Howard G. Tucker and Ralph P. Boas (Auth.)
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Extra resources for An Introduction to Probability and Mathematical Statistics
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1] F ,, X ABSOLUTELY CONTINUOUS DISTRIBUTIONS . . 5. We now prove the condition is necessary. Assuming the random variables to be independent we find that for every (x\, • • •, xn) n By the very definition of a joint density, the integrand of the last multiple integral above is a joint density, and the theorem is proved. Conditional distributions are needed in much work in probability and statistics, and the remainder of this section is devoted to these. Definition. The conditional distribution function of a random variable X given that a random variable Y = y is defined by Fx\v{x\y) = lim P([X ^ x] \ [y - e < Y ^ y + e]), «-*o+ provided that such a limit exists.
Yes, it is, in a practical sense. Table 1 should convince the student of this. The explanation of the table is easy. This is data collected by Rutherford and Geiger. 5 seconds of duration. , they observed the number of alpha particles emitted during 2608 disjoint intervals. The first column, n, of Table 1 lists the number of par- 26 DEPENDENT AND INDEPENDENT EVENTS [Chap. 5) with X = . 5-second time interval, namely, n = 0, 1, 2, • • • . 5-second intervals exactly one alpha particle was emitted, in 383 intervals two particles were emitted, etc.
4. Let / / \ /O if min {z, z/} < 0 Find Fx Ax, y), *x(x), FY(y)Jx(x), 2 2 and/rfo). 2 5. Let /x,r(z, I/) = l/7r (l + z )(l + 2/ ) for all real x, problem 4.