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Your computations are almost correct, there is only a subtle error. In order to have $XY\leq e^t$, given that $X\geq 1$ we must have $Y\leq e^t$. Therefore, the correct integral should be $$F_Z(t)= \... • 8,699 1 vote ### Constructing a density function from a condtion Write your functional equation as$$ f(x) = \frac{1/\nu - \theta h f(xh)}{1-\theta}$$You have f(x) = 0 for x > \nu. Thus for \nu \ge x > \nu/h you must have f(x) = 1/(\nu (1-\theta)). ... • 453k 0 votes ### Related to CDF of product of two independent Gamma random variables It's quite hard to get an explicit formula for CDF of a product of Gamma r.v.s. One of the ways is to use the Mellin transform, which has a great condition, that if you have 2 random variables, then ... 0 votes Accepted ### What does the values of Joint Probabilities indicate here? It simply means$$f(x,y) = \begin{cases} \frac{1}{2}, & 0 \le y \le x \le 1 \\ \frac{3}{2}, & 1 < x \le x+y \le 2 \\ 0, & \text{otherwise}. \end{cases}$$It's not actually clear what ... • 140k 0 votes Accepted ### Solving a stochastic equation by characteristic functions I got it. We have to use the Mellin-transform mentioned in the Wikipedia article about the distribution of the product of two random variables. The Wikipedia-article quotes the property, that for ... 0 votes ### Calculate probability density function of a random vector (X + Z, Y + W) The form of the density implies that the conditional distribution of Y given X is that of a normal distribution with mean X and variance X, and that the marginal distribution of X is that of ... • 1,568 0 votes ### Density function of the sum of 2 random variables In case you are not up on Jacobian transformations, you could consider the joint cdf for X+Y and Y.$$ F_{X+Y,Y}(u,v) = P(X+Y \leq u, Y \leq v) $$You can compute this by integrating over the ... 2 votes ### Switching limits of a pdf The region is a parallelogram. You only need one integral if you integrate with respect to y first. For each x, y goes from x to x+1 ,so the integral should be$$ \int_{0}^{1} \int_{x}^{x+1} ...
For a question like this, it's almost always a good idea to sketch the region in the $xy$ plane. Now if you look at how lines of constant $y$ cut this parallelogram, you see that you need different ...