1. If X has an exponential distribution with theparameter θ, use thedistribution function technique to find the probability density of the randomvariable Y=In X.2. If the joint probability density of X and Y isgiven by f(x,y) = 4xyexp(-(x^2 + y^2)) for x>0, y>0 and Z=sqrt(X^2 + Y^2), find(a) The distribution function of Z;(b) The probability density of Z.3. If n independent random variables have normal distributions with the means and the standard deviations , find the moment-generatingfunction of their sum and identify the corresponding distribution, its mean,and its variance. 4. If the number of minutes that a doctor spendswith a patient is a random variable having an exponential distribution with theparameter θ = 9,what are the probabilities that it will take the doctor at least 20 minutes totreat (a) one patient; (b) two patient? 5. If the probability density of X is given byf(x)= x/2 for 0 <2, find the probability density of Y=X^3. 6. Let X and Y be two continuous random variables having thejoint probability density f(x,y) = 4 for 0 <1 and 0< y<1., find the jointprobability density of Z=X^2 and Z'=XY
更新1:
For #1, my answer is 1-exp(-e^y/θ), but im not sure what is the range of y. I dont think it is y>0.
#2a. I have 1-(z-1)exp(-z^2) for z>0
2b. I have (1-2z^2+2z)(-exp(-z^2))
更新2:
There is one more question that i dont really get it: If X is the total we roll with a pair of dice, find the probability distribution of the remainder we get when the values of X divided by 3. (The probability distribution is given)
