The joint probability density function of is a function such that for any hyper-rectangle. The notation used in the definition above has the following meaning:. Furthermore, the notation where is a -dimensional vector is used interchangeably with the notation where are the entries of. Finally, the notation means that the multiple integral is computed along all the co-ordinates. Let be a random vector having joint pdf. In other words, the joint pdf is equal to if both components of the vector belong to the interval and it is equal to otherwise.
Suppose we need to compute the probability that both components will be less than or equal to. This probability can be computed as a double integral:. Instead, it has the PDF illustrated in Exhibit 3. A random vector is joint-normal with uncorrelated components if and only if the components are independent normal random variables.
A property of joint-normal distributions is the fact that marginal distributions and conditional distributions are either normal if they are univariate or joint-normal if they are multivariate. Select k components. Without loss of generality, suppose these are the first k components X 1 , X 2 , … X k. Let X 1 be a k -dimensional vector comprising these components, and let X 2 be an n — k -dimensional vector of the remaining components.
This generalizes property [ 3. Next, one thinks about the number of ways of selecting two white and two red balls. One does this in steps — first select the white balls, then select the red balls, and then select the one remaining black ball. Note that five balls are selected, so exactly one of the balls must be black. Here the probability of choosing a specific number of white and red balls has been found.
To do this calculation for other outcomes, it is convenient to define two random variables. The table of probabilities is given in Table 6. It is clear from Table 6. Using Table 6. The variable is a vector containing the colors of the ten balls in the box.
The function simulates drawing five balls from the box and computing the number of red balls and number of white balls. Using the replicate function, one simulates this sampling process times, storing the outcomes in the data frame results with variable names X and Y.
Using the table function, one classifies all outcomes with respect to the two variables. By dividing the observed counts by the number of simulations, one obtains approximate probabilities similar to the exact probabilities shown in Table 6. The marginal pmf is displayed in Table 6. Note that a marginal pmf is a legitimate probability function in that the values are nonnegative and the probabilities sum to one.
Table 6. This conditional pmf is just like any other probability distribution in that the values are nonnegative and they sum to one. Recall that the data frame results contains the simulated outcomes for selections of balls from the box. Note that the relative frequencies displayed below are approximately equal to the exact probabilities shown in Table 6. Suppose one rolls the usual six-sided die where one side shows 1, two sides show 2, and three sides show 3.
This situation resembles the coin-tossing experiment described in Chapter 4. One is repeating the same process, that is rolling the die, repeated times, and one regards the individual die results as independent outcomes.
The difference is that the coin-tossing experiment had only two possible outcomes on a single trial, and here there are three outcomes on a single die roll, 1, 2, and 3. This formula can be used to compute a probability for our example. Other probabilities can be found by summing the joint Multinomial pmf over sets of interest.
One attractive feature of the Multinomial distribution is that the marginal distributions have familiar functional forms. A more intuitive way to obtain a marginal distribution relies on the previous knowledge of Binomial distributions.
In each die roll, suppose one records if one gets a one or not. One applies the knowledge about marginal distributions to compute conditional distributions in the Multinomial situation. An alternative way to figure out the conditional distribution is based on an intuitive argument. Using the replicate function, one simulates the Multinomial experiment for iterations. The outcomes are placed in a data frame with variable names X1 , X2 and X3. Given this simulated output, one can compute many different probabilities of interest.
Figure 6. This region of points in the plane is shown in Figure 6. One can check that the pdf in our example is indeed a legitimate pdf.
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