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In Spivak's Calculus 3rd Edition, there is an exercise to prove the following: $$x^n - y^n = (x-y)(x^{n-1} + x^{n-2} y + ... + x y^{n-2} + y^{n-1})$$ I can't seem to get the answer. Either I've g...
For discrete random variables, the conditional probability mass function of Y Y given the occurrence of the value x x of X X can be written according to its definition as. P (Y = y \mid X = x) = \dfrac {P (X=x \cap Y=y)} {P (X=x)}. P (Y = y ∣ X = x) = P (X = x)P (X = x∩Y = y).
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Let X and Y be two discrete random variables. The joint PMF of X and. Y is defined as. pX,Y (x, y) = P[X = x and Y = y]. (1) Figure: A joint PMF for a pair of discrete random variables consists of an array of impulses. To measure the size of the event A, we sum all the impulses inside A.
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Discrete random variables X1, X2, …, Xn are independent if the joint pmf factors into a product of the marginal pmf's: p(x1, x2, …, xn) = pX1(x1) ⋅ pX2(x2)⋯pXn(xn). It is equivalent to check that this condition holds for the cumulative distribution functions.
Suppose that X and Y are jointly distributed continuous random variables with joint pdf f(x, y). If g(X, Y) is a function of these two random variables, then its expected value is given by the following: E[g(X, Y)] = ∬ R2g(x, y)f(x, y)dxdy. We will give an example applying Theorem 5.2.1 in an example below.
