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Lecture 18

Lecture 18. On Normal random variables. Review: 1D pdf. f X (x). x. a. b. Review: 2D pdf. y. R. x. Review: independence. y. If X and Y are independent, then f X,Y (x,y) = f X (x)f Y (y). d. R. c. Justification. a. b. x. R: X+Y  z. Review: Sum of 2 random variables.

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Lecture 18

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  1. Lecture 18 On Normal random variables

  2. Review: 1D pdf fX(x) x a b

  3. Review: 2D pdf y R x

  4. Review: independence y If X and Y are independent, then fX,Y(x,y) = fX(x)fY(y). d R c Justification a b x

  5. R: X+Yz Review: Sum of 2 random variables Z =X+Y. X and Y may or may not be independent. What is the cdf of Z? y x

  6. Review: Sum of 2 random variables Z =X+Y. X and Y may or may not be independent. What is the pdf of Z?

  7. Review: Sum of 2 random variables Z =X+Y. X and Y are independent. What is the pdf of Z? Convolution of fX and fY

  8. Example • Two objects O1 and O2 with unknown weights W1 and W2. • Weigh O1 and O2. Precision: 0.5 gram. • Measurements M1 and M2: • W1 = M1+e1. • W2 = M2+e2. • e1 and e2 are errors. • Sum of weights is M1 + M2 1.

  9. Example (cont’d) • Model the errors e1 and e2 by uniform distribution. • e1 and e1 are independent. Each of them is evenly distributed between –0.5 and 0.5. • E[ei] = 0, Var(ei)=1/12. • Pdf of total error e1 + e2 is triangular. • Pdf of sum of weights is also triangular, centered at M1+M2.

  10. Sum of many uniform r.v.’s • What is the error when we sum the measurement of n objects? • The true total weight is between M1+M2+…+ Mn n/2. • For j = 1,…,n, E[Wj]=Mj. Var[Wj]=1/12. • E[W1+W2+…+ Wn]=M1+M2+…+ Mn. • Var(W1+W2+…+ Wn)=n/12.

  11. When n is large… • Distribution of W1+W2+…+Wn is approximately equal to normal distribution with mean M1+M2+…+Mn and variance n/12. • This is a consequence of the central limit theorem.

  12. Recreation: Weighing design • There is a clever way that measure each weight with less variance. • A = W1+W2+e1. • B = W2+W3+e2. • C = W1+W3+e3. • E[(A+C–B)/2]= W1, • Var((A+C–B)/2)=(3/4)(1/12) is reduced by a factor of 3/4.

  13. Standard normal • A zero-mean unit-variance normal (or Gaussian) random variable is called a standard normal random variable. • Transformation from normal to standard normal • Z is normal with mean  and variance 2. • (Z – )/ is standard normal.

  14. Function of two indep. normal • X and Y are independent normal random variables. • For simplicity, suppose they are zero-mean, and Var(X) = 2=Var(Y). • X+Y is normal. • X2+Y2 is exponential. • Sqrt(X2+Y2) is Rayleigh.

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