The commom probability density of the $X$ and $Y$ RVs are as follow: $$f_X(x)=  \begin{cases}     e^{-x}&amp; \text{if } x&gt; 0\\     0,              &amp; \text{otherwise} \end{cases}$$ The question is to estimate the PDF of $XY$. My attempt: $$Z = XY$$ $$f_Z(z) = \int_{-\infty}^{\infty}\frac{1}{|x|}f(x, {z \over x})dx$$ Since given RVs are iid in nature, hence $$f_Z(z) = \int_{-\infty}^{\infty}\frac{1}{|x|}f(x)f({z \over x})dx$$ $$f_Z(z) = \int_{0}^{\infty}\frac{1}{x}\exp\bigg(-x-{z \over x}\bigg)dx$$ Now I'm stuck at this point. How can I solve this integral?
I am trying to define the probability distribution of $Z$ such as $Z = X_1\cdot X_2$ where $X_1$ and $X_2$ are two independent and identically exponentially distributed variables.  $$P(X_1=x) = \lambda e^{-\lambda x}$$ $$P(X_2=x) = \lambda e^{-\lambda x}$$  I tried something like that….  $$P(Z=x) = P(X_1 = a) \cdot P(X_2 = x/a)$$  Is this first equation correct?  $$P(Z=x) = \lambda e^{-\lambda a} \cdot \lambda e^{-\lambda x/a} = \lambda^2 e^{-\lambda (a+x/a)}$$  I doesn't feel like I found the solution as the dummy variable $a$ remains in the final result. Can you please help me solving this problem?
My internet connection is very slow. I suspect someone in the network is using a BitTorrent client which takes huge amount of bandwidth.  Is there a way to detect who is using my bandwidth (network sniffing tools etc)?  I should have pointed out that I have no access to the router, what I am looking for is a way of detecting someone's MAC/IP if she uses Bittorrent. What is $n!$ when $n=0$? using $$ in a post causes strange number series to appear Take a look at this example schematic:  The chip in question is the  and as you can see power is provided on both sides (with a 100nF cap to smooth noise I guess). Is this strictly required or could you put power into one side and just ignore the other? I know that I have just put power in one side and everything still seems to work fine but I feel uneasy with doing that. I feel that they would not waste pins that could be doing something useful by having an extra pair of power in/out pins. For what purpose does this PIC MCU (and others like it) have multiple power pins? I am a UI designer and developer and have been designing websites for aound 10 years. I want to get into user experience design. I know how to do a design, but I am lagging the terminologies and the technical things in the user experience field.   So how do I go about getting the skills and mindset necessary for being a User Experience Designer? Can I attack a specific player? I was in a clan and someone kicked me from it.   I think I'm stronger than him. Is there any way I can attack him? The only method I see to attack is random players. Can I attack a specific player? I realize this is probably going to come off as a bit silly, but in Iron Man II how could James Rhodes easily don the Mark II armor that was designed to fit Tony Stark?  As you can see in this image (which isn't an "in universe" shot) Tony and Rhodey are roughly the same height, but it appears that Rhodey has a smaller frame than Tony, which would seem to make for an ill fit of the armor, especially for flight.  This issue could of course have been addressed by the Air Force during the retrofit with Justin Hammer. Overstaying on a US Visa Waiver Program A friend of mine came from Australia on the WVP and stayed 92 days. I made a mistake booking Her flight scheduling a Saturday arrival/departure at DFW. This amounted to a 92 day stay and with no funds to rebook a low cost flight through Qantas it was necessary to leave on the scheduled flight.   Question # 1) Was a automatic I-94W submitted by the airlines? She had no customs forms to fill out upon departure.   Question #2) How does this effect another WVP being issued if applied for? Deep learning model (LSTM) with temporal and non temporal attributes Identical coefficients estimated in Poisson vs Quasi-Poisson model When is $\sin(x)$ rational? 16.04 new installation gives grub-efi-amd64-signed failed installation /target/ ubuntu 16.04 at the end Is JavaScript's "new" keyword considered harmful? In another , a user pointed out that the new keyword was dangerous to use and proposed a solution to object creation that did not use new. I didn't believe that was true, mostly because I've used Prototype, Scriptaculous and other excellent JavaScript libraries, and everyone of them used the new keyword.  In spite of that, yesterday I was watching Douglas Crockford's talk at YUI theater and he said the exactly same thing, that he didn't use the new keyword anymore in his code ().  Is it 'bad' to use the new keyword? What are the advantages and disadvantages of using it? What is cool about generics, why use them? I thought I'd offer this softball to whomever would like to hit it out of the park. What are generics, what are the advantages of generics, why, where, how should I use them? Please keep it fairly basic. Thanks. Relationship between gamma and chi-squared distribution If $$Y=\sum_{i=1}^{N}X_i^2$$ where $X_i \sim \mathcal{N}(0,\sigma^2)$, i.e. all $X_i$ are i.i.d. normal random variables of zero mean with same variances, then $$Y \sim \Gamma\left(\frac{N}{2},2\sigma^2\right).$$  I know the chi-squared distribution is a special case of the gamma distribution, but could not derive the chi-squared distribution for random variable $Y$. Any help, please? Is there any rule for when it is better to use genitive case or noun as adjective? I'm not sure if there is any difference in meaning in this example:     The department of accounting       The accounting department   Can I also use both forms in situations like:     The restaurants of Paris / The Paris restaurants      The room window / The window of the room let $n\gt1$  be a fixed natural number, $S:=${$A $: $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$   $d(A,B)$:=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and $GL(n,\mathbb R)$ is set of all $n\times n$ non singular matrix how to prove that $GL(n,\mathbb R)$ is  not connected and open subset of $S$ ?  thanks in advance When a user is suspended, their account is reduced to 1 rep and all of his privileges are stripped. However, they can still join chat rooms.   This leads to various forms of possible abuse (all of which I've seen by suspended users):   Join/leave spams Expletives in profile messages Hold one sided conversations with users of the room via    Suspended users cannot be kicked, and even moderators can do very little (aside from completely removing the account, which is often overkill).  My proposal is to extend  for suspended users to one of two levels:   Prevent them from being shown in the user list at all. Strip their profile profile picture and description from the profile view visible from chat rooms (not from profile pages). We've upgraded some machines to Windows 10 and realized there were some updates which updated as required. However, I realized there was no option available to stop the download similar to that on Windows 7 and 8.1. The only way I could stop the download was to stop the Windows Update service.  My question is does anyone know of a way to stop auto updates or is stopping the service is the only solution? How to get horizontal black line when coloring cells?