Question Let $X$ and $Y$ be the number of accidents which will occur at each of two intersections over the next year. Suppose that $X$ and $Y$ are independent Poisson random variables , with means $a$ and $b$ respectively. Find the conditional distribution of the number of accidents which will occur at the first intersection over the next year, given the total number of accidents. My working Let $W = X + Y$ $\implies f_W(w) = \frac {(a + b)^w e^{-(a + b)}} {w!}$ $f_{X \mid W}(x \mid w) = \frac {f_{X, W}(x, w)} {f_W(w)}$  This is where I am stuck. I believe that, in order to find the conditional distribution of $X$ on $W$, I need the joint distribution of $X$ and $W$ - I know of no other way to approach the problem. However, I am not sure if I am given enough information to find this joint distribution. In particular, it is obvious that $X$ and $W$ are not independent. How should I continue? Any intuitive explanations will be greatly appreciated :)
The classic example... $X \sim Po\left (\lambda\right ), Y \sim Po\left (\mu\right)$, X and Y are independent.  Show that the conditional distribution of X is binomially distributed. Or in other words, $P(X=k\mid X+Y = n) = P (\tilde{X} = k), \tilde{X} \sim B\left (n ,\frac{\lambda}{\lambda + \mu}\right )$.   I've so far managed to reach to this step, and have been stuck since. Just somehow gotta get a $\frac{1}{n!}$ in the denominator, that would then complete the proof..or at least I think..  $$P(X=k\mid X+Y=n) = \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{P(X+Y = n)}=  \frac{\frac{\lambda^{k}\mu^{n-k}}{k!(n-k)!}}{\sum_{i=1}^{n} \frac{\lambda^{i}\mu^{n-i}}{i!(n-i)!}}$$  Thanks for the help!
sed fails to remove newline character Efficient unsigned-to-signed cast avoiding implementation-defined behavior Let's assume a very simplified situation. There is only one seller asking for 10$ and there is only one buyer bidding 20$. At what price would the transaction go?   Some answers say it won't, because the ask price has to match the bid price. () Other answers say, that the time ordering is important. () Which answer is correct? What is the final price in the presented situation? Every non-empty subset of the integers which is bounded above has a largest element. I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof:  Since $S$ is a non-empty subset of $\mathbb{Z}$(hence $\mathbb{R}$) which is bounded above, by the supremum property $sup S$ exists. Let $w=supS$, then we have to show that $w \in S$, so we suppose $w \notin S$, then we have $w-1&lt;w$, then exists $m\in S$ such that $w-1&lt;m&lt;w$(here is my problem, why can we take $m\in S$ such that happens? we are in $\mathbb{Z}$, so that does not happens). Then he takes $n\in S$ such that $w-1&lt;m&lt;n&lt;w$. The inequality $n&lt;w$ implies that $-w&lt;-n$, and $w-1&lt;m$, if we add up both inequalities we have $-1&lt;m-n$ implies $n-m&lt;1$, then $0&lt;n-m&lt;1$ which is a contradiction because there are no integers between $0$ and $1$ Can anybody explains me if that step is right?? How can I prove this if there is a problem with that step? Thanks! Is it correct to say "I kindly request you to..."? Isn't kindness already implied when you say "I request you to..."?  When I say "I humbly request you to...", the word humbly helps me to label the state of my behavior during the request.  On the other hand, what value does the word kindly add to the request? How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution.  This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving the continuity of $\sin$, but I can't find out how. Any help is appreciated. Is there a fast way (without looping over all edges/faces/verts) using ops.mesh or bmesh to get the lists of indices associated with mesh islands as shown below? These islands are part of one mesh but are disjoint from each other.    I know how to select the boundaries by running the operator Select boundary Loop or calling bpy.ops.mesh.region_to_loop() but can't find any API calls to return the vert/edge/face indices associated with the individual islands. The new profile shouldn't mistakenly imply that you can earn the Informed badge on a meta site This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless  Imagine there are a 100 people in line to board a plane that seats 100. The first person in line realizes he lost his boarding pass so when he boards he decides to take a random seat instead. Every person that boards the plane after him will either take their "proper" seat, or if that seat is taken, a random seat instead.  Question: What is the probability that the last person that boards will end up in his/her proper seat.  Moreover, and this is the part I'm still pondering about. Can you think of a physical system that would follow this combinatorial statistics? Maybe a spin wave function in a crystal etc... How should a bot earn enough reputation to perform the actions necessary for that bot? Recovering lost text which was copy pasted from a website to MS Word If $A$ is a square matrix and $A^2 = 0$ then $A=0$. Is this true? If not, provide a counter-example. I want to create a text file which will add each command I am going to run on the terminal as a new line entry in the text file. i.e., I need to keep a record of what are the things I'm doing in the terminal. Further if I can keep a record of outputs I'm getting in the terminal, along with the inputs in the terminal, as a text file, it will be much better.   Each time I forget a command I have to search again in the Internet for the command.  Can I can make a file like above? Is it possible to have several different views of one scene and set them as named "presets"? What happens to the energy when waves perfectly cancel each other? What happens to the energy when waves perfectly cancel each other (destructive interference)? It appears that the energy "disappear" but the law of conservation of energy states that it can't be destructed. My guess is that the kinetic energy is transformed into potential energy. Or maybe it depends on the context of the waves where do the energy goes? Can someone elaborate on that or correct me if I'm wrong? Can I run Android apps on Ubuntu? Is it possible to run Android apps on Ubuntu? And I don't mean inside an emulator, but natively, as you would with any other application installed using the Ubuntu Software Center. How do I find the common ratio of a geometric sequence? A geometric sequence has its first term equal to $12$ and its fourth term equal to $-96$.   How do I find the common ratio? And find the sum of the first $14$ terms How to make many copies of similar graphs systematically? I want to make various copies of a graph in tikz. The graph I want has 10 vertices. But only has the edges between the vertices a_i,a_(i+1),t,b.  I would like to automatically create 8 copies of this graph. and draw them. I just don't know how to code it. Here is my current code, but it adds all of the edges to the same copy of the graph:  \begin{tikzpicture} [main_node/.style={circle,fill=blue!20,draw,minimum size=2em,inner sep=3pt]}] \node[main_node] (8) at (0:0.5) {b}; \node[main_node] (9) at (180:0.5) {t}; \foreach \x in {0,1,...,7}     \node[main_node] (\x) at (360/8*\x:2) {$a_{\x}$}; \foreach \x in {0,1,...,7}{     \pgfmathsetmacro{\y}{int(mod(\x+1,8))}     \path[draw,thick] (\x) edge node {} (8);     \path[draw,thick] (\x) edge node {} (9);     \path[draw,thick] (\x) edge node {} (\y);     \path[draw,thick] (8) edge node {} (9);     \path[draw,thick] (8) edge node {} (\y);     \path[draw,thick] (9) edge node {} (\y);    }   I want each of the edge sets to be drawn seperately. How to prevent US post code formats I have an issue with American and Canada postcodes in accounts and contacts. Some of the users are creating the salesforce records with postcodes like (MD 20904), when we sending post code from salesforce to SAP integration, SAP system is rejecting salesforce records as postcode is not in correct format.   My question is:   what are actual US and canada post code formats? If Users are creating the post codes with MD 20904, how would prevent through validation rule like only accept sepecific format? Union of ascending chain of Topologies