The joint probability mass function (joint pmf), or, simply the joint distribution, of two discrete r.v. X and Y is defined as p(x,y) = P(X = x,Y = y) = P({X = x}∩{Y = y}).

In this chapter we consider two or more random variables defined on the same sample space and discuss how to model the probability distribution of the random variables jointly. We will begin with the discrete case by looking at the joint probability mass function for …

The joint probability distribution can be expressed in terms of a joint cumulative distribution function and either in terms of a joint probability density function (in the case of continuous variables) or joint probability mass function (in the case of discrete variables).

Description of multivariate distributions • Discrete Random vector. The joint distribution of (X,Y) can be described by the joint probability function {pij} such that pij. = P(X = xi,Y = yj). We should have pij ≥ 0 and X i X j pij = 1.

Joint probability mass functions: discrete random variables. I. If X and Y assume values in {1, 2,..., n} then we can view A. i,j = P{X = i, Y = j} as the entries of an n × n matrix. I. Let’s say I don’t care about Y . I just want to know P{X = i}. How do I figure that out from the matrix? P n. I Answer: P{X = i} = j=1 A i,j.

Joint Probability Distributions Definition: (a) The joint distribution of X and Y (both discrete) is defined by p(x;y) = P(X = x;Y = y) satisfying (i) p(x;y) 0; (ii) P x;y p(x;y) = 1: (b) Also, p (x) =P X = x X y p(x;y); p (y P(Y = y) = X x p x;y) are respectively called the marginal distributions of X and Y: (c) The mean (or the expected ...

Figure 5‐1 Joint probability distribution of X and Y. The table cells are the probabilities. Observe that more bars relate to less repeating. For a discrete joint PDF, there are marginal distributions for each random variable, formed by summing the joint PMF over the other variable.

The joint pdf \[ f(x, y) = \begin{cases} \text{dnorm(x) * dexp(y)} & y > 0\\ 0 & \text{otherwise} \end{cases} \] describes independent random variables, because the region over which the random variables are positive is \((-\infty, \infty) \times [0, \infty)\) and the function factors as dnorm(x) times dexp(y).

Joint probability distributions may also be defined for n random variables. If. Independence of random variables X and Y implies that their joint CDF factors into the product of the marginal CDFs. assuming that f(x,y) exists.

9.2 Random vectors and joint distributions Recall that a random variable X is a real-valued function on the sample space (Ω,F,P), where P is a probability measure on Ω; and that it induces a probability measure PX on R, called the distribution of X, given by PX(I) = P(X ∈ I) = P {ω ∈ Ω : X(ω) ∈ I}, for every interval in R. The ...