CS 530: Geometric and Probabilistic Methods inComputer ScienceHomework 1 (Fall ’06)1. The joint p.m.f. of two discrete random variables X and Y is given below:YX5/12 1/18 5/721/36 1/36 5/361/12 1/6 1/72.Determine whether X and Y are statistically independent.2. The joint p.m.f. of two discrete random variables X and Y is given below:YX0 1/9 1/31/18 5/36 1/121/36 1/4 0.Compute and tabulate:(a) Marginal p.m.f., pX(xi).(b) Marginal p.m.f., pY(yj).(c) Conditional p.m.f., pX|Y(xi|yj).(d) Conditional p.m.f., pY |X(yj|xi).3. Prove thatnr=nn − r.4. After years of driving through a particular intersection, you know that theprobability of being stopped by a red light is 0.27. You drive through theintersection twice a day, five days a week. What is the probability of beingstopped by a red light five or more times in a single week?5. Let X be a discrete random variable with the following p.m.f.:pX(xi) =c x2ixi∈ {1,2,3,4,5}0 otherwise.• Find c.• Find the expected value and variance of X.6. The continuous random variable X has p.d.f.:fx(x) =c x2if 1 ≤ x ≤ 50 otherwise.• Find c.• Find the expected value and variance of X.• Find the probability that X exceeds 2.0.7. A p.d.f. is defined as follows:fX(x) =2x/9 if 0 ≤ x ≤ 30 otherwise.Find the value of FX(1), i.e., the value of the c.d.f. at 1.8. Let X and Y be continuous r.v.’s wherefX(x) =1τe−x/τif t > 00 otherwise.and let Y = X2. Derive an expression for fY.9. During World War II, German propaganda claimed that they could targettheir flying bombs, and most of the British public believed this to be true. Toinvestigate this claim, we divide the area of London into 576 equal areas of1/4 square kilometers each. The table shows the number of areas receivingdifferent numbers of hits. First, calculate the average number of hits perarea. Second, on the assumption that the bombs cannot be aimed with anyaccuracy, calculate the number of areas receiving k = 0,1,2,3,4, and 5 hits.k 0 1 2 3 4 5# of areas with k hits 229 211 93 35 7 110. Two fair dice are rolled, one of which is red and the other is green. Let W bethe difference (red number less green number). Compute and plot pW(w).11. Write a MATLAB function which computes the 2D joint histogram, GXY,of a pair of images, X and Y , of equal size. Test it on the red and greencomponents of the Queen Butterfly Fish image found on the class home-page. Display the joint histogram, GXY, as a grey level image. [Note: Usethe online documentation to find out how to generate hardcopies of imagesin MATLAB (or Octave).]12. Write a MATLAB function which, given a joint histogram, GXY, returnsthe marginal histograms, GXand GY. Using the Queen Butterfly Fish im-age, verify that the marginal histograms you compute are the same as thosecomputed using the 1D histogram function found on the class homepage.[Note: Use the online documentation to find out how to plot and generatehardcopies of histograms in MATLAB (or
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