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Lectures prepared by: Elchanan Mossel Yelena Shvets

Learn how to compute the distribution of Z using different operations on random variables. Understand the methods for finding the joint density, CDF, and convolution formula.

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Lectures prepared by: Elchanan Mossel Yelena Shvets

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  1. Lectures prepared by:Elchanan MosselYelena Shvets

  2. Operations on Random Variables Question: How to compute the distribution of Z = f(X,Y)? Examples:Z=min(X,Y),Z = max(X,Y), Z=X+Y, . Answer 1: Given the joint density of X and Y, f(X,Y), We can calculate the CDF of Z, fZ(z) by integrating over the appropriate subsets of the plane. Recall: If X & Y are independent the joint density is the product of individual densities: f(x,y) = fX(x) fY(y), However, in general, it is not enough to know the individual densities.

  3. Distribution of Z = X+Y Discrete case: Continuous case: Therefore: And: For independent X & Y:

  4. Distribution of Z = X+Y – using CDF

  5. Sum of Independent Exponentials Suppose that T & U are independent exponential variables with rate l. Let S = T + U, then fS(s) is given by the convolution formula:

  6. Sum of Independent Uniform(0,1) • If X,Y » Unif(0,1) and Z = X + Y then 2 1 0 2 0

  7. Sum of Independent Uniform(0,1) • If X,Y,W » Unif(0,1) and T = X + Y+W then T = Z + W, 0<t<1 1<t<2 2<t<3 2 2 2 1 1 1 0 0 0 0 0 0 2 1 2 1 2 1

  8. Sum of Three Independent Uniform(0,1) The density is symmetric about t=3/2. 0 2 1 3

  9. Example: Round-off errors Problem: Suppose three numbers are computed, each with a round-off error Unif(-10-6,10-6) independently. What is the probability that the sum of the rounded numbers differs from the true sum by more than 2£ 10-6? Solution: x1+ R1 x2+ R2 x3+ R3 Ri» Unif(-10-6, 10-6) x1+ R1+x2 + R2+x3 + R3=x1+x2 +x3 +( R1+ R2+ R3) Want: p = 1 - P(-2 £ 10-6 < R1+ R2+ R3< 2 £ 10-6 ) Let: Ui = (Ri /10-6 + 1)/2. Ui ~ Unif(0,1) are independent. p = 1 - P(1/2 < U1 + U2 + U3 < 5/2) = 2P(T>5/2) = 2/48.

  10. Ratios Let Z = Y/X, then for z>0, the event Z 2 dz is shaded. For independent X & Y:

  11. Ratio of independent normal variables • Suppose that X & Y»N(0,s), and independent. Question: Find the distribution of X/Y. We may assume that s =1, since X/Y = X/s / Y/s. This is Cauchy distribution.

  12. Distribution of Z = Y/X – using CDF

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