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- Joint distributions and independence
- 5.2: Joint Distributions of Continuous Random Variables
- Joint probability density function

Bivariate Rand. A discrete bivariate distribution represents the joint probability distribution of a pair of random variables. For discrete random variables with a finite number of values, this bivariate distribution can be displayed in a table of m rows and n columns. Each row in the table represents a value of one of the random variables call it X and each column represents a value of the other random variable call it Y.

Each of the mn row-column intersections represents a combination of an X-value together with a Y-value. The numbers in the cells are the joint probabilities of the x and y values. Notice that the sum of all probabilities in this table is 1. Since f x,y is a probability distribution, it must sum to 1. Adding probabilities across the rows you get the probability distribution of random variable X called the marginal distribution of X.

Adding probabilities down the columns you get the probability distribution of random variable Y called the marginal distribution of Y.

The next display shows these marginal distributions. The main property of a discrete joint probability distribution can be stated as the sum of all non-zero probabilities is 1. The next line shows this as a formula. The marginal distribution of X can be found by summing across the rows of the joint probability density table, and the marginal distribution of Y can be found by summing down the columns of the joint probability density table. The next two lines express these two statements as formulas.

A continuous bivariate joint density function defines the probability distribution for a pair of random variables. The graph of the density function is shown next. For the case of the the joint density function shown above, integration amounts to finding volumes above regions in the xy plane. A bivariate continuous density function satisfies two conditions that are analogous to those satisfied by a bivariate discrete density function.

First f x,y is nonnegative for all x and y, and second. Also, like the bivariate discrete case, marginal continuous densities for random variables X and Y can be defined as follows:. In the discrete case conditional probabilities are found by restricting attention to rows or columns of the joint probability table.

For example, the table below shows the joint probabilities of random variables X and Y defined above. The marginal probabilities are also shown. Any conditional probability for a pair of discrete random variables can be found in the same way. The technique shown above for a conditional probability in the discrete case doesn't carry over to the continuous case because the 'row' the probability of a specific value of X and 'column' the probability of a specific value of Y totals are zero in the continuous case.

In fact, the joint probability of a specific value of X and a specific value of Y is zero. The approach taken to get around this limitation is to define conditional probability density functions as follows:. An example of a conditional density computation comes from exercise 5. To avoid subscripts, the example will be done here using X in place of X 1 and Y in place of X 2. From exercise 5. The region over which this density function is nonzero is shown within the triangle below the x axis is horizontal and the y axis is vertical.

This definition of independence for discrete random variables translates into the statement that X and Y are independent if and only if a cell value is the product of the row total times the column total. Are the random variables X and Y described above with the following joint probability density table independent? The marginal density functions can be multiplied together to produce the joint density function.

Thus the random variables X and Y are independent. The following two formulas are used to find the expected value of a function g of random variables X and Y. The first formula is used when X and Y are discrete random variables with pdf f x,y. The next formula is used when X and Y are continuous random variables with pdf f x,y.

In computing E[X - Y] for the random variables X and Y whose joint pdf is 1 for x in [0,1] and y in [0,1] and 0 otherwise, you get the following.

The covariance is a measure of association between values of two variables. If as variable X increases in value, variable Y also increases, the covariance of X and Y will be positive. If as variable X increases in value, variable Y decreases, the covariance of X and Y will be negative.

If as X increases, there is no pattern to corresponding Y-values, the covariance of X and Y will be close to zero. The covariance of X and Y are defined as follows. Using covariance to measure the degree of association between two random variables is flawed by the fact that covariance values are not restricted to a real number interval. This flaw is overcome by using the correlation coefficient for two random variables. The correlation coefficient is a normalized form of covariance whose values are restricted to the interval [-1,1].

Thus these two random variables have a weak positive association. Finally, a result for computing expected values and variances for linear combinations of random variables. The following statements show how expected values and variances of these linear combinations are computed.

It was also assumed that outcomes on any run of the experiment were independent. Random variables considered under these assumptions were the Bernoulli, Binomial, Geometric, and Negative Binomial, and since the Poisson is a limiting form of a Binomial, in some sense, the Poisson. The multinomial random variable generalizes the situation described in the first paragraph by allowing more than one two outcomes on each run of the experiment.

If you think of tossing a coin as the model for the random variables described in the last paragraph, tossing a die is a good model for the multinomial random variable. If you toss the coin n times, you might want to record the number of 1's, the number of 2's, etc.

You can simulate multinomial random variables on a computer by dividing the interval [0,1] into k subintervals where k is the number of different possible outcomes. For example to simulate the tossing of a die, have the computer generate a uniform random variable on [0,1]. If the number falls into the first subinterval, a 1 has been tossed, if the number falls into the second subinterval, a 2 has been tossed, etc.

Probability Distribution The probability distribution of the multinomial with parameters n, p 1 ,p 2 ,p 3 ,p 4 ,p 5 ,p 6 is. Y Values.

When introducing the topic of random variables, we noted that the two types — discrete and continuous — require different approaches. The equivalent quantity for a continuous random variable, not surprisingly, involves an integral rather than a sum. Several of the points made when the mean was introduced for discrete random variables apply to the case of continuous random variables, with appropriate modification. Recall that mean is a measure of 'central location' of a random variable. An important consequence of this is that the mean of any symmetric random variable continuous or discrete is always on the axis of symmetry of the distribution; for a continuous random variable, this means the axis of symmetry of the pdf. The module Discrete probability distributions gives formulas for the mean and variance of a linear transformation of a discrete random variable.

These ideas are unified in the concept of a random variable which is a numerical summary of random outcomes. Random variables can be discrete or continuous. A basic function to draw random samples from a specified set of elements is the function sample , see? We can use it to simulate the random outcome of a dice roll. The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value. For the dice roll, the probability distribution and the cumulative probability distribution are summarized in Table 2. We can easily plot both functions using R.

Bivariate Rand. A discrete bivariate distribution represents the joint probability distribution of a pair of random variables. For discrete random variables with a finite number of values, this bivariate distribution can be displayed in a table of m rows and n columns. Each row in the table represents a value of one of the random variables call it X and each column represents a value of the other random variable call it Y. Each of the mn row-column intersections represents a combination of an X-value together with a Y-value. The numbers in the cells are the joint probabilities of the x and y values.

*In probability theory and statistics , the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset.*

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Having considered the discrete case, we now look at joint distributions for continuous random variables. The first two conditions in Definition 5. The third condition indicates how to use a joint pdf to calculate probabilities.

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