MLE for special uniform case I'm preparing for my exam and I came across this question:      Let $X_1, X_2, ...,X_n$ be iid with PDF $f(x)=\frac{2x}{\theta^2}$ for $0 \leq x \leq \theta.$ Find the MLE of $\theta$   So this is what I did:   $L(\theta)= \prod\frac{2x_i}{\theta^2}I(0 \leq x \leq \theta)$  $= (2n \bar{x})(\frac{1}{\theta^{2n}})I(0 \leq min (x_i)\leq max(x_i) \leq \theta) $  $=(2n \bar{x})(\frac{1}{\theta^{2n}})I(0 \leq X_{(1)}\leq X_{(n)} \leq \theta)$  In order to find MLE, I need to maximize $L(\theta)$, by choosing a small $\theta$. However, $\theta$ can't be smaller than $ X_n$. Thus I conclude that:  $\hat{\theta}= max(x_i)$  Is my calculation and conclusion correct?
MLE of $\theta$ when $X_1,\ldots,X_n $ are i.i.d with $P_\theta(x; \theta)=2x/\theta^2$ for $0\le x \le\theta$ Assume the standard situation, that is, let $X_1, \ldots , X_n$ be independent and identically distributed with $X_k ∼ P_\theta(x; \theta)$ , where $P_\theta(x; \theta) = 2x/\theta^2$  if $0\le x \le\theta$ and $0$ otherwise. It is required to estimate $\theta$. Show that the maximum likelihood estimator for $\theta$ is $\hat{\theta} = \max{[X_1, \ldots , X_n]}$ and then show that the cumulative distribution function of $\hat{\theta}$ is $F_\theta(z) = z^{2n}/\theta^{2n}$. Here's what i did so far: Maximum Likehood estimator: $L_x(θ) = \prod_{i = 1}^{n} P_θ(x_k)$ Here we have $P_θ(x_1,...,x_n;θ) = P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)$ Likehood = $L_{x,θ}(θ) =P_θ(x_1,θ).P_θ(x_2,θ)...P_θ(x_n,θ)= 2x_1/θ^2. 2x_2/θ^2.. 2x_n/θ^2 = [2^n.\prod_{i = 1}^{n}x_i]/θ^{2n}$ Log-likehood: $\sum_{i = 1}^{n}log(P_θ(x_1,...x_n;θ))= \sum_{i = 1}^{n}log(2x_i/θ^2)$ Is this correct so far? Im still not sure how to get to $\hat{θ} = max{[X_1, . . . , X_n]}$ As for the the cumulative distribution part to show $F_θ(z) = z^{2n}/θ^{2n}$: $F(z) = P(max({x_k})&lt;z) = P(x_1&lt;z).P(x_2&lt;z)..P(x_n&lt;z) = 2x_1/θ^2.2x_2/θ^2...2x_n/θ^n = 2^n.\prod_{i = 1}^{n}x_i/θ^2n$ Not sure if this is correct. Would really appreciate some help. Edit: From the answers below, we can deduce the the estimator is biased. What estimator would be unbiased? How can i find it?
For example Lets use the Marrutact from Sandstorm, it can cast as a 5th level wizard. if it then takes a prestige class that requires the ability to cast 1st level Arcane spells and advances arcane casting such as Abjurant Champion. assuming it meets all other prerequisites does the Marrutacts spellcasting improve when it takes a level in Abjurant Champion or does it gain no benefit? Prove that a subset of a separable set is itself separable Extracting numbers from vectors of strings How does the coin limit work? The new gym update , including a limit of 50 coins earned per day. How does this limit work? Is each pokemon limited to accruing 50 coins per day? Are all my pokemon limited to a collective 50 per day? My teammates keep feeding my Dragonite and she has been at the gym for almost a week; will she return with 50 coins for each day she was there? Fewer than 50 for the days I had other pokemon in the gym? Or will she accumulate vast sums of coins over the week, but only deliver me 50 (or fewer) when she comes back based on what I have collected from returning pokemon that day? How can we rewrite this optimization problem in standard form? Consider the following problem $$ \begin{array}{crl} &amp;\underset{y}{\max} &amp; f(y)\\ s.t. &amp; y_{1}A_{1}+y_{2}A_{2}+S_{1}&amp;=C_{1},\\  &amp; y_{3}A_{3}+y_{2}A_{4}+y_{5}A_{5}+S_{2}&amp;=C_{2},\\  &amp; y_{3}A_{6}+y_{4}A_{7}+S_{3}&amp;=C_{3},\\  &amp; S_{1},S_{2},S_{3}\succeq0, \end{array} $$ where $y_{i}$ are scalar variables and the matrices $A_{i}$ are symmetric real. How can we rewrite this optimzation problem in the following form  $$ \begin{array}{crl} &amp;\underset{y}{\max} &amp; f(y)\\ s.t. &amp; \sum_{i=1}^{m}y_{i}A_{i}+S&amp;=C,\\   &amp;S&amp;\succeq0, \end{array} $$ where $x_{i}$ are scalar variables and the matrices $B_{i}$ are symmetric real. please? Thanks. How can I get exactly 5 logarithmic divisions of an interval? Should my flair be including sites with less than 200 reputation? Is it ok to use an alternate account while my main user is restricted? I'm new to Stack Exchange, and my main user only has 16+ rep. And it's been blocked in Code Review. I need to use Code Review today. Should I use an alternate account? I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. Is "Call me when you are available to talk" correct? Call me when you are available to talk   Is that correct? Can I use it that way? Does the Dueling Fighting Style prevent using a shield? The Dueling fighting style doesn't seem to state anything about shields, stating that the character can only wield one weapon while fighting to gain the benefit. Does a shield count as a weapon or otherwise stop the damage bonus from occurring? JavaScript get clipboard data on paste event (Cross browser) How was Thor able to dent Captain America's Shield? I was reading about Adamantium on the      Proto-Adamantium   This is the original Adamantium created by Dr. MacLain. Proto-Adamantium is the official name for the metal that makes up Captain America's Shield. It has never been recreated and is the absolute most invulnerable item in all of existence.    This lead me to read about      The true shield was once slightly dented by an enraged Odinforce empowered Thor, using Mjolnir, but he later reconciled and pounded out the small dent.    Was the dent caused by Odinforce or Thor's Hammer? Is Mjolnir or Captain America's Shield "the absolute most invulnerable item in all of existence"? How to prevent my screens from turning off, when the screen is locked? I would like to have my screen fully powered at all times, but it goes to standby once I lock my PC. Using ArcPy to convert .txt to shapefile with coordinate system? I tried to define my coordinate system on creating the shapefile but when I zoom to layer I don't see my features and ArcGIS crashes. I know it's not in the coordinate system I defined for it but it shows the coordinate under properties but doesn't reflect when I zoom to layer. Here is my python code:  import arcpy  newfcName ="newfc.shp"   outpath = r"C:/FLOOD/IKEJA AND KOSOFE"  # Declaration arcpy.env.overwriteOutput=True  arcpy.env.workspace= outpath  # Create new Shapefile and add FIELDS newfc = arcpy.CreateFeatureclass_management(outpath, newfcName, "Point") # get the coordinate system by describing a feature class dsc = arcpy.Describe("newfc.shp" )  coord_sys = dsc.spatialReference  try:        # run the tool    arcpy.DefineProjection_management(newfcName , coord_sys)     # print messages when the tool runs successfully    print(arcpy.GetMessages(0))  except arcpy.ExecuteError:    print(arcpy.GetMessages(2))  except Exception as ex:    print(ex.args[0])  arcpy.AddField_management(newfc, "X", "STRING", field_length = 50) arcpy.AddField_management(newfc, "Y", "STRING", field_length = 50) arcpy.AddField_management(newfc, "Z", "STRING", field_length = 50) arcpy.AddField_management(newfc, "I", "FLOAT", field_length = 50)  # Reference Cursors cursor=arcpy.da.InsertCursor(newfc, ["X", "Y", "Z", "I"])  # Read File  a = open("C:/FLOOD/IKEJA AND KOSOFE/DEM/532722.txt","r")  inputF = a.readlines()  for line in inputF:     xCoordinate, yCoordinate, zValue, iValue = line.split(" ")      newRow = (str(xCoordinate), str(yCoordinate), str(zValue), float(iValue))      cursor.insertRow(newRow)  a.close() How can I have one keyboard input layout and multiples languages in Ubuntu 17.10? If I have English language I can't type "'e" and get an "é", as I can do in Portuguese and Spanish.  When I manage to do that on Settings -> Region &amp; Language, I'm able to change languages, but it also changes the keyboard layout.  Is there any way to change language (Portuguese or Spanish for example) and set the keyboard input layout (US International for example)? VLC will not install 14.04 I tried to watch a dvd and found that VLC was not installed. When I went to the Ubuntu software center I got this message when trying to install VLC. I do not know why. Here's the ouput:  "This error could be caused by required additional software packages which are missing or not installable. Furthermore there could be a conflict between software packages which are not allowed to be installed at the same time.  The following packages have unmet dependencies:  vlc: Depends: vlc-nox (= 2.1.4-0ubuntu14.04.1) but 2.1.4-0ubuntu14.04.1 is to be installed      Depends: libc6 (&gt;= 2.15) but 2.19-0ubuntu6.4 is to be installed      Depends: libfreetype6 (&gt;= 2.2.1) but 2.5.2-1ubuntu2.2 is to be installed      Depends: libgcc1 (&gt;= 1:4.1.1) but 1:4.9.1-0ubuntu1 is to be installed      Depends: libqtcore4 (&gt;= 4:4.8.0) but 4:4.8.5+git192-g085f851+dfsg-2ubuntu4 is to be installed      Depends: libqtgui4 (&gt;= 4:4.8.0) but 4:4.8.5+git192-g085f851+dfsg-2ubuntu4 is to be installed      Depends: libstdc++6 (&gt;= 4.6) but 4.8.2-19ubuntu1 is to be installed      Depends: zlib1g (&gt;= 1:1.2.3.3) but 1:1.2.8.dfsg-1ubuntu1 is to be installed"   Thanks in advance for any help. How do I play a YouTube video in the background on my iPad? Where can I see how many posts have I edited? Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$.