How to show $X_n$ converges in distribution to U(0,1) Assuming $X_n$ has a discrete distribution on the set $\{\frac{1}{n}\frac{2}{n},...,\frac{n-1}{n},1\}$. We want to show that $X_n \xrightarrow{D}U(0,1)$. I tried to find the CDF of $X_n$ to start, but I unsure how to do this. If I find $\lim_{n\to\infty}F_{X_N}$ and then compute the partial derivative with respect to $t$ and show this is equivalent to $U(0,1)$ I think I will be done. But I don't know how to find the CDF. Any help with this would be greatly appreciated. Thanks
Convergence to a Uniform Distribution $\newcommand{\floor}[1]{\left\lfloor #1 \right\rfloor} $  Show that if $P(X_n = i/n)=1/n$ for every $i = 1,...,n$, then $X_n$ converges in distribution to a uniformly distributed random variable $X$.  Convergence in distribution is defined as   $$P(X_n \leq x) \rightarrow P(X \leq x)$$  and I have that a uniform distribution with parameters $a$ and $b$ implies that  $$P(X \leq x) = \frac{x-a}{b-a}$$  I cannot go any further however. I feel like I don't know anything about $X_n$ - I don't even know if it is continuous or discrete. What I can just say is that   $$P(X_n \leq x) = \sum_{i=1}^\floor{x} \frac{1}{n}$$  where $\floor{x}$  means rounding $x$ to the closest lower integer.
Is it normal for focal length to change while focusing? When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change.  More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance.  This is most noticeable at the 55mm end of the lens.  Is this normal? Encountering a FileSystemRights value that isn't defined in enumeration Is it appropriate to buy a "thank you" gift for a PhD supervisor? I am nearing the end of my PhD and I would quite like to buy a "thank you" gift for my supervisor (although, she's the socially awkward type who I'm sure will be greatly embarrassed to receive one).  I'm just wondering whether a gift is the done thing? And if so, what sort of gift seems appropriate? I suppose a bottle of wine would be a safe choice? Vote for Tag Synonyms Directly on the Page Before:  After:   The pros:  Speeds up the tag synonym voting process. Easier to locate (as opposed to navigating to each individual tag wiki page). Encourages users to vote on more than one at a time.  The cons:  Serial up/downvoting. But this really isn't even an issue because, (1) only experienced users can vote on tag synonyms, and (2) more than one vote is required to approve a tag synonym, so the occasional serial up/downvoter won't significantly impact the overall process. Upgrading from a modern smartphone camera without going all in on a DSLR kit Android SQLite database: slow insertion When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? How do I drill a 2⅛ inch door knob hole over an existing 1½ inch hole? I need to replace the door knob for this interior door:    The door knobs I've found at the stores require a 2⅛ inch hole.  Mine is only 1½:    Did door knobs used to be smaller?  Is there some place I can find a replacement door knob that will fit my door?  If I need to modify the door to fit a modern door knob, will the new hole completely overlap the old? and can I reuse the latch hole? Use of comma separating two adjectives Using the validation data I'm unclear on the exact process of using the validation data.  Let's say that I fit my neural network model and adjust hyperparameters using the training set and validation set. Do I then evaluate the test set on this model? Or do I recombine the validation and training sets and fit a fresh model with the hyperparameters that I found during the validation phase, and then evaluate on the test data? I have seen a number of different notebooks and examples that do both ways.  Surely, once I've found my hyperparameters, it makes sense to fit a fresh model using the full training set (recombined with validation set), since the validation loss has no effect on the weights. 'bridges' on a PCB copper layer I've come across a design where each pad was connected using 4 'bridges' to the GND copper layar. What stands behind these 'bridges'? Why not make a full copper layer with only solder-mask defining the pads? I can't retrieve my Facebook password I haven't been to my Facebook account in a while and I can't remember my password. When I try to have it reset, it goes to a security question, but I can't remember that either.    Is there anything I can do? What causes an electric shock - Current or Voltage? Though voltage and current are two interdependent physical quantity, I would like to know what gives more "shock" to a person - Voltage or Current? In simple words, will it cause more "electric - shock" when the voltage is high or Current is more? What happens if a Rainbow Servant takes levels in Mystic Theurge? Enroll Apple Developer Program under 18 I've seen a lot of questions like this throughout the web but they're mostly from 2 or 3 years ago, so I decided to ask to have some updated answers.   I'm 16 and I want to start publishing and using the benefits that comes with the Developer Program. In the legal agreement says that I should be over 18, and I'm not so, i can't do it. I've contacted the apple developer support and they told me that I should use a account in my father's name. We reside in Brazil. With that, I had some doubts:   Will the name on the App Store be mine or my father's? In the future, when I'm 18, this account can be transferred to my name?   Besides that, if someone has any advice or (legal) workaround I'd appreciate it, thanks Proof for triangle inequality for vectors Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below.  By considering $u$ and $v$ as two sides of a triangle, we can see that the lengths of the third side is $\| u + v \|$ and we have $\| u + v \| \leq \|u\| + \|v\|$.  Under what circumstance equality occurs and how can one prove that? What is the probability that all three tosses result in the same outcome? A coin is tossed three times: The probability of zero heads is 1/8 and the probability of zero tails is 1/8.  And my question is: What is the probability that all three tosses result in the same outcome?   So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?  If so, we could use the Addition Rule which is $P(A \cup B) = P(A) + P(B)$  where A and B are disjoint events, i.e. $A \cap B = \emptyset$, A is the event of tossing all heads and B is the event of tossing all tails.  I'm not sure how to continue after that... would the complement be used? Delete specific line number(s) from a text file using sed? I want to delete one or more specific line numbers from a file. How would I do this using sed? Don't invite people when moving their messages Why would Louise telling the Chinese commander about his wife's last words change his mind? In the future in Arrival, the commander says that he changed his mind about going to war with the aliens because Louise called his private number and told him his wife's dying words. My question is what did Louise say to him (like in English) and why did it change his mind?