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When are two random variables x and Y independent?
Two random variables X and Y are independent if any pair of events of the form X∈A, Y∈B are independent. For discrete random variables, it is enough to show that Pr [X=x ∧ Y=y] = Pr [X=x]⋅Pr [Y=y], or in other words that the events [X=x] and [Y=y] are independent for all values x and y.
Which is the definition of the expectation of XY?
Expectation of XY: the definition of E(XY) Suppose we have two random variables, X and Y. These might be independent, in which case the value of X has no effect on the value of Y. Alternatively, X and Y might be dependent: when we observe a random value for X, it might influence the random values of Y that we are most likely to observe. For
Which is the correct distribution for random variable x?
Here are some common distributions for a random variable X: Pr [X = 1] = p, Pr [X = 0] = q, where p is a parameter of the distribution and q = 1-p. This corresponds to a single biased coin-flip. Pr [X = k] = (n choose k) p k q (n-k), where n and p are parameters of the distribution and q = 1-p.
When do we expect a random variable to be positive?
The mean, expected value, or expectation of a random variable X is writ- (X,Y) will be positive if large values of X tend to occur with large values of Y, and small values of X tend to occur with small values of Y. For example, if X is height and Y is weight of a randomly selected person, we would expect cov(X,Y) to be positive. 50
What is the bound for the correlation of three random variables?
The variance of any linear combination of x, y, z must be non-negative. Let the variances of these variables be σ2, τ2, and υ2, respectively. All are nonzero (for otherwise some of the correlations would not be defined).
What kind of function is a random variable?
Figure 1: A (real-valued) random variable is a function mapping a probability space into the real line. As such, a random variable has a probability distribution. We usually do not care about the underlying probability space, and just talk about the random variable itself, but it is good to know the full formalism.