Identification of the Distribution of Random Variables
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University of California, Santa Barbara. Retrieved April 26, The Practice of Statistics 2nd ed. New York: Freeman. Archived from the original on Dharmaraja Introduction to Probability and Stochastic Processes with Applications. Tsitsiklis, John N. Belmont, Mass. Fristedt, Bert; Gray, Lawrence A modern approach to probability theory. Kallenberg, Olav Random Measures 4th ed. Berlin: Akademie Verlag. Foundations of Modern Probability 2nd ed. Berlin: Springer Verlag. Papoulis, Athanasios Probability, Random Variables, and Stochastic Processes 9th ed.
Tokyo: McGraw—Hill. Outline Index. Descriptive statistics. Mean arithmetic geometric harmonic Median Mode. Central limit theorem Moments Skewness Kurtosis L-moments. Index of dispersion.
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Example b average of 50 rolls shows the same range 1 through 6 of outcomes as Example a individual rolls , but Example b has more possible outcomes. You could get an average of 3. Also, the shape of the graphs are different; example a shows a flat, uniform shape, where each outcome is equally likely, and Example b has a mound shape; that is, outcomes near the center 3. This is to be expected. If you were to roll a die 50 times, you would expect the average to be near the average of the values 1,2,3,4,5,6 since each of those values are equally likely to occur.
The average of 1,2,3,4,5,6 is 3.
- Random variables.
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We are interested in determining the probability law of the travel distance for the ambulette to reach a random medical emergency. Solution Following the general discussion above, a derived distribution problem is like any other probabilistic modeling problem; it requires that we do four things to model the experiment: STEP 1: Define the random variables of interest.
STEP 2: Identify the joint sample space. STEP3: Determine the joint probability distribution over the sample space. STEP 4: Work within the sample space to determine the answers to any questions about the experiment. As discussed above, the activity specific to derived distributions functions of random variables occurs in Step 4. Random variables.
Suppose that the highway is of unit length. Joint sample space. The joint sample space is the unit square in the positive quadrant 0 X 1 1, 0 X 2 1. Joint probability distribution.
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We will assume that the locations of the ambulette and the medical emergency are uniformly, independently distributed over the highway. In practice, the three assumptions entailed in such a statement would have to be argued for plausibility and measurements might have to be taken. Naturally, the analysis could also proceed with an alternative set of assumptions.
Since we are now dealing with strictly continuous random variables, we will work with the joint probability density function, which is 4. Work in the sample space. This is the point at which the never-fail method for deriving distributions comes into play. Consideration of these two cases gives rise to the shaded region in the sample space in Figure 3. This is often the most difficult part of a derived distribution problem. Note that determination of this region in no way depended on the joint pdf for X 1 and X 2 ; thus, the "work" invested to this point could be applied to several alternative models, each with its own joint pdf for X 1 and X 2.
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Since the joint X 1 , X 2 pdf is uniform over the unit square, we can perform the integration by computing areas in the sample space. Conceptually, each area is multiplied by "l," the height of the pdf at that point, to yield a probability measured as a volume. By computing areas of the triangles not in the shaded region, we have now completed step b of the never-fail method and we are "done.
For instance, the expected value or mean value of D is These results will be of use in our further work.
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A system administrator may be interested in knowing the effects on travel distance of prepositioning the ambulette at the center of the interval depicting the highway, thus fixing X 2 Then the joint sample space is the straight line indicated in Figure 3. Thus, the pdf of D'is How could this result also be obtained by inspection? The mean and variance are Thus, a change in deployment policy resulting in an ambulette prepositioned at the center of its service area rather than randomly patrolling its service area reduces mean travel distance by 25 percent, the variance of the travel distance by Question: How would one determine or estimate the joint distribution function for X 1 and X 2 in practice?
Further work: Problems 3. Extension: Scaling We often select the scale of a probabilistic modeling problem for analytical convenience. For instance, if the length of highway analyzed in Example I had been Thus, after performing the analysis for a conveniently scaled problem, we often rescale it to suit the real-world situation at hand. Scaling can also occur when switching systems of measurement, say from British units to metric units.
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Suppose that we have derived the probability law for W, given one scale, and we wish to find the moments and the probability law of In words, multiplying a random variable by a constant results in its variance being multiplied by the square of that constant. We can also derive the probability law of V assumed to be continuous using the never-fail method.
These equations constitute the answer to our problem. Returning to the patrolling ambulette example, the cdf for X 1 becomes You might find it helpful to sketch several different applications of this result. Exercise 3. The location of the medical emergency X 1 , Y 1 and of the ambulette X 2 , Y 2 are independently uniformly distributed over the response area. Instead, we may be concerned with the rightmost coordinate R and the leftmost coordinate L. For instance all points between R and L may be exposed to siren and lights as the ambulette passes at high speed.
Thus, the joint probability law of R and L would be of interest. We will ignore scaling and assume that all locations, as before, occur in the interval [0, 1]. Solution Since we have already performed Steps in describing the experiment, we are ready to go to Step 4 work in the sample space and employ the neverfail method. The random variables that are functions of the original random variables are We wish to derive the joint probability law for R and L.
To execute step a of the never fail method, we proceed formally as follows: To proceed from here, we consider separately each of the two events in braces and "merge" these later by intersection. Combining these two cases, the event Max X 1 , X 2 r corresponds to the square of area r 2 shown in Figure 3. Combining these two cases, the event Min X 1 , X 2 1 corresponds to the L-shaped region shown in Figure 3.