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5. LetX, Ybe independent exponential random variables with mean 1. Find the jointdensity of (X+Y, X-Y). Write down clearly the ranges for the variables.
6. Let the joint probability density ofX, Yis given byfX,Y(x, y) =(Cxyif0< x < y <10otherwise.FindCand the marginal density ofX. Also findP(2X≤Y).RyR1013dy=18. SoC= 8.7. Suppose that the expected number of accidents per week at an industrial plant is 5.Suppose also that the numbers of workers injured in each accident are independentrandom variables with a common mean of 3. If the number of workers injured in eachaccident is independent of the number of accidents that occur, compute the expectednumber of workers injured in a week.8. A fair coin is tossed 100 times, and supposeYdenotes the total number of timesheads turned up. LetXdenote the number of tails among the first 50 tosses. FindV ar(Y|X= 0). (Hint: Check ifXandY-50 +Xare independent).
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Probability,Probability theory,heads,joint probability density