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Lecture 21

Lecture 21. Probability density function of jointly normal random variables. Single variable. X~N(,  2 ). Last time. X , Y are independent standard normal. U = aX+bY V = cX+dY Want to find the distribution of (U,V). Notation for multivariable. Mean vector. Covariance matrix.

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Lecture 21

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  1. Lecture 21 Probability density function of jointly normal random variables

  2. Single variable • X~N(, 2)

  3. Last time • X, Y are independent standard normal. • U = aX+bY • V = cX+dY • Want to find the distribution of (U,V)

  4. Notation for multivariable Mean vector Covariance matrix

  5. Covariance of U and V • X, Y are independent standard normal. • U = aX+bY • V = cX+dY • Var(U) = a2+b2, Var(V) = c2+d2, Cov(U,V)=ac+bc. • Covariance matrix of U and V is

  6. Numerical example • X, Y are independent standard normal. • U = 2X+Y • V = X+2Y

  7. 2D normal distribution Note: If K is diagonal, it reduces to product of two normal pdf. Important implication: If two jointly normal random variables are uncorrelated, then they are independent.

  8. Some Linear algebra • A is a square matrix. •  is an eigenvalue of A and v (0) is an eigenvector of A if Av= v. • A is diagonalizable if it is real and symmetric. We can find two orthonormal vectors v1 and v2,

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