Contents
What is the product of two random variables?
Say X1, X2, …, Xn are independent and identically distributed uniform random variables on the interval (0, 1). What is the product distribution of two of such random variables, e.g., Z2 = X1 ⋅ X2?
Which is the probability product of two uniform distributions?
In general, we can conjecture that fZn(z) = { ( − logz)n − 1 ( n − 1)!, 0 < z ≤ 1 0, otherwise, which we can prove via induction on n. I leave this as an exercise. If X1 is uniform, then − logX1 ∼ Exp(1).
Which is true when two random variables are statistically independent?
When two random variables are statistically independent, the expectation of their product is the product of their expectations. This can be proved from the Law of total expectation : In the inner expression, Y is a constant. Hence: This is true even if X and Y are statistically dependent. However, in general is a function of Y.
Which is a type of algebra for random variables?
The product is one type of algebra for random variables: Related to the product distribution are the ratio distribution, sum distribution (see List of convolutions of probability distributions) and difference distribution. More generally, one may talk of combinations of sums, differences, products and ratios.
A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product.
What is the variance of X and Y?
where Var(X + Y) is the variance of the sum of X and Y, Var(X – Y) is the variance of the difference between X and Y, Var(X) is the variance of X, and Var(Y) is the variance of Y. Note: The standard deviation (SD) is always equal to the square root of the variance (Var).
What is an independent quantity?
The following are true of an independent quantity: It describes the independent variable. It can also be described as the input values in a functional relationship. It is represented by the x–coordinate in the ordered pair (x, y) in a functional relationship. It determines the value of the related dependent quantity.