计量经济学 Multivariate Distribution

That is, the probabilities assigned to the joint outcomes are nonnegative.
2. Σ Σ f(x, y) = 1.
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2.3 Cumulative Joint Probability Function A function F(x, y) that gives the probability that the random variable X is less than a given value xi and the random variable Y is jointly less than a given value yi: F(x, y) = P(X≤x and Y≤y) = Σs≤x Σt≤y f(s, t). is known as the joint distribution function or the cumulative joint probability function.
Lecture 5: Multivariate Distributions
1. Introduction In the previous two lectures we considered the probability distribution for a single random variable. We then learned how to use various measures (e.g., mean and variance) to summarize the information in the distribution. We have, however, limited ourselves to looking at a single random variable (e.g., X).
That is, we are often interested in multivariate distributions. In this lecture we extend our analyses to include two random variables (i.e., bivariate or joint distributions). This is a straightforward extension of the previous two chapters but, importantly, with two random variables being considered we have the additional scope of looking at relationships between the two variables. The results developed for bivariate case are easily extended beyond the bivariate situation. We will start with the case where both of the random variables are discrete.
2. F(1,3) = P(X ≤ 1 and Y ≤ 3) = 1.
This must be the case as it covers all possible outcomes for X and Y.
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2.4 Bivariate and Univariate Distributions Sometimes, when given the bivariate distribution for two random variables X and Y, we would like to move back to the univariate distributions for X and Y. We call these the marginal distributions for X and Y. The marginal distributions for X and Y are, respectively, given by
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Example 1. Continuing with the coin-toss example above, we could calculate the probability of X being less than or equal to 0 and Y (jointly) being less than or equal to 2. This would simply be sum of the probabilities satisfying both of 3 conditions (i.e., X ≤ 0 and Y ≤ 2). Thus, F(0,2) = P(X ≤ 0 and Y ≤ 2) = 1/8 + 1/4 + 1/8 + 0 = ½.
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Notice that we indicate the different values X can assume in the left column of the table (e.g., 0,1) and the different values Y can assume in the top row of the table (e.g., 0,1,2,3).
Notice the similarity of the joint distribution function and the cumulative probability distribution for a univariate distribution.
Note also that the calculation simply involves summing all the probabilities associated with outcomes satisfying X≤ x and Y≤ y.
Thus, we have been considering univariate distributions.
Often in economics we are interested in relationships existing between several random variables.
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2. Discrete Bivariate Distributions 2.1 Joint Probability Function Suppose we have two discrete random variables, X and Y, which each take on a finite set of values.
In this case the table is a matrix and the graph of the distribution would be 3 dimensional.
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Example Continuing with the above example, consider the outcome (X=0, Y=0). This joint outcome occurs in one of the 8 outcomes in the sample space (i.e., TTT). Thus, P(X=0 and Y=0)=1/8. Similarly, the joint outcome (X=0, Y=1) occurs in 2 of the 8 basic outcomes (i.e., HTT, THT). Thus, P(X=0 and Y=1)=1/4. Proceeding in a similar fashion we can calculate the probability associated with each joint outcome. It is helpful to depict this information in a table (matrix) as shown in Table 5.1.
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Example Suppose we flip a coin 3 times and record the number of heads. We will define the random variable X to equal the number of heads on the last (third) flip, and the random variable Y to equal the total number of heads in three flips. Then, S ={HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, and the random variable X takes on the values {0, 1} while the random variable Y takes on the values {0, 1, 2, 3}. There are 8 possible different joint outcomes: (X = 0, Y = 0), (X = 0, Y = 1), (X = 0, Y = 2), (X = 0, Y = 3), (X = 1, Y = 0), (X = 1, Y = 1), (X = 1, Y = 2), (X = 1, Y = 3). If we attach a probability to each of the different joint outcomes, we have a discrete bivariate probability distribution or, more formally, a joint probability function. 4
We assume X takes on the values {x1, x2, x3, … xN} and Y takes on the values {y1, y2, y3,…, yN}.