Joint, marginal and conditional distributions, covariance, correlation and sums of random variables.
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- Define a joint probability mass function (joint PMF) for two discrete random variables X and Y.
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For each pair of values (x, y), the joint PMF is P(X = x, Y = y), giving the probability that X equals x AND Y equals y simultaneously.- Define a joint probability density function (joint PDF) for two continuous random variables X and Y.
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A function f(x, y) such that P(a < X < b, c < Y < d) equals the double integral of f(x, y) over the region [a, b] x [c, d].- How do you obtain the marginal PMF of X from a joint PMF P(X = x, Y = y)?
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Sum over all possible values of Y: P(X = x) = sum of P(X = x, Y = y) for all y.- How do you obtain the marginal PDF of X from a joint PDF f(x, y)?
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Integrate over all possible values of Y: f_X(x) = integral of f(x, y) dy from negative infinity to infinity.- Define the conditional PMF of X given that Y = y.
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P(X = x | Y = y) = P(X = x, Y = y) / P(Y = y), provided P(Y = y) > 0.- Define the conditional PDF of X given that Y = y.
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f(x | y) = f(x, y) / f_Y(y), provided f_Y(y) > 0.- State the condition for X and Y to be independent random variables in terms of joint and marginal distributions.
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X and Y are independent if and only if P(X = x, Y = y) = P(X = x) * P(Y = y) for all x, y (or f(x, y) = f_X(x) * f_Y(y) in the continuous case).- Define the covariance of two random variables X and Y.
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Cov(X, Y) = E[(X - E[X])(Y - E[Y])].- State the alternative formula for covariance in terms of products of random variables.
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Cov(X, Y) = E[XY] - E[X] * E[Y].- Define the correlation coefficient between X and Y.
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Corr(X, Y) = Cov(X, Y) / (sqrt(Var(X)) * sqrt(Var(Y))) = Cov(X, Y) / (sigma_X * sigma_Y).- What is the range of the correlation coefficient?
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The correlation coefficient is always between -1 and 1, inclusive: -1 <= Corr(X, Y) <= 1.- If X and Y are independent, what is Cov(X, Y)?
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Cov(X, Y) = 0.- Can two random variables with zero covariance be dependent?
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Yes, zero covariance does not imply independence; they can have a nonlinear dependence.- State the formula for Cov(aX + b, cY + d) where a, b, c, d are constants.
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Cov(aX + b, cY + d) = ac * Cov(X, Y).- If X and Y are independent, what is Corr(X, Y)?
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Corr(X, Y) = 0.
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