1 / 10

Lectures prepared by: Elchanan Mossel Yelena Shvets

Lectures prepared by: Elchanan Mossel Yelena Shvets. 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, .

hedia
Download Presentation

Lectures prepared by: Elchanan Mossel Yelena Shvets

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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. 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:

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

  6. 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

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

  8. 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.

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

  10. 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.

More Related