$X$ and $Y$ are independent Poisson distributed values, means are $θ$ and $2θ$.  Consider the combined estimator $$\gamma = k_1\cdot X + k_2\cdot Y$$ where $k_1$ and $k_2$ are arbitrary constants.  (a) Find the condition on $k_1$ and $k_2$ such that $\gamma$ is an unbiased estimator.  To solve this I let $\theta = E[k_1X + k_2Y] = k_1E[X] + k_2E[Y]$  $\theta = k_1\theta + 2k_2\theta$, $k_1 + 2k_2 = 1$  However I'm stuck when it comes to the next 2 parts:  (b) For $\gamma$ unbiased, show that the variance of the estimator is minimized by taking $k_1 = 1/3$ and $k_2 = 1/3$.  (c) Given observations x and y find the maximum likelihood estimate of θ. and hence show that $\gamma$ is also the maximum likelihood estimator.  I'm not sure where to even start for part (b) and part (c). Any help would be appreciated!
Suppose that X and Y are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde{\theta} = k_1 X + k_2 Y $$ where $k_1$ and $k_2$ are arbitrary constants.   Find the condition on $k_1$ and $k_2$ such that $\tilde{\theta}$ is an unbiased estimator of $\theta$. For $\tilde{\theta}$ unbiased, show that the variance of the estimator is minimized by taking $k_1 = 1/3$ and $k_2 = 1/3$. Given observations $x$ and $y$ find the maximum likelihood estimate of $\theta$ and hence show that $\tilde{\theta}$ is also the maximum likelihood estimator.   I have gotten (1) and (2) okay, but it's (3) I am having trouble with, I'd be okay if $X$ and $Y$ had the same parameter but I'm having trouble with $X$ and $Y$ having different parameters, any help would be appreciated.   NOTE   For (1) I got $k_1 = 1 - 2k_2$.   For (2) I found the variance of $\tilde{\theta}$, then differentiated and let equal to zero to minimize - therefore we get (after subbing in $k_2 = 1 - k_1/2 $) $$3k_1-1=0,$$ which when subbing in $1/3$, we see it is minimised.   Thank you.
Which should I use on my DB? - Clustered Index or Non Clustered Index or Both I just need advice on how I am going to optimize my database.  Date ---- Time ---- Area ---- Block ---- Data1 ---- Data2 ---- Data3 ---- DataN   I have that format of tables on my database. Each tables has 30-days of records with hundred thousands of records each data. Uploading of data to database is every morning. The data of yesterday will be uploaded.  The most common queries to run is grouped by Date or Date and Area or Date and Area and Block. Now, in order to make queries faster, I found out using indexes. I used indexes before. But that was when I am using a primary key. In this table, I didn't use primary key because it is totally unnecessary.   Now, I am really confused on how I am going to optimize this. Based on  on Column Considerations part, clustered index is not a good choice on Columns that undergo frequent changes. I am really confused.  Please help me if I need to use clustered index here or just non clustered or both. Am not doing anything fancy, just trying to update a field on Contact in JS of LWC but getting this exception.  Uncaught TypeError: 'set' on proxy: trap returned falsish for property 'Name' throws at mydomain/auraFW/javascript/mhontaYdOya4Y_lBu7v9yg/aura_prod.js:2:27687  HTML Code:   &lt;template&gt;      &lt;template if:true={wiredContact}&gt;          {wiredContact.Name}          &lt;lightning-input value={wiredContact.Name} onchange={updateName}&gt;&lt;/lightning-input&gt;     &lt;/template&gt;  &lt;/template&gt;   JS:   import { LightningElement ,wire,track,api } from 'lwc'; import myContact from "@salesforce/apex/ContactController.fetchContact";  export default class Myrefreshapextest extends LightningElement {    @track wiredContact;   @wire (myContact)         fetchedContact({error, data}){             if(data){                 console.log(JSON.stringify(data));                 this.wiredContact = data;             }else if (error){                 console.log(error);             }     }      updateName (event){         console.log(JSON.stringify(event.detail.value));         console.log(JSON.stringify(this.wiredContact));         this.wiredContact.Name = event.detail.value;     }  }   Apex:  public class ContactController {      @AuraEnabled(cacheable=true)     public static Contact fetchContact(){         return [SELECT Id,Name FROM COntact LIMIT  1];     } }   On top of my head, am not doing anything wrong, anyone has idea what's wrong with my code?  When I print console.log(JSON.stringify(this.wiredContact)); I get old values so am pretty sure it exists.  I tried with @track and @api, but same response. Can anyone shed some light? Roman numerals become “too wide” in table of contents With the ls command, is it possible to show only the files created after a specific date, hour...?  I'm asking it because I have a directory with thousand of files.  I want so see all files that were created since yesterday.   I use ls -ltr but I have to wait to see all files...  There is an equivalent of DIRECTORY/SINCE=date from OpenVMS ? Windows Unable to Delete ._. File A virtual file containing the concatenation of other files How can we disable triggers, Validation Rules and workflow rules in salesforce org. [on off - switch] How can we disable triggers, Validation Rules and workflow rules in salesforce org.  This becomes really needed and important when we are doing data migration(data imports) in salesforce.  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Particularly, I've been trying to prove that $id_A$ (the identity morphism on $A$, for all $A \in Obj(\mathbf{Set})$) is $1_A \colon A \rightarrow A, x \mapsto x$.  This is a very intuitive and reasonable statement, and it's trivial to prove that $1_A$ is indeed an identity morphism on $A$, and I suppose uniqueness of $id_A$ can be demonstrated analogously to uniqueness of the identity element in a monoid (considering the subcategory which has $A$ as its only object, and endofunctions on $A$ as its only morphisms).  In this manner, it is not hard to prove that the proposition in the title is true, but this demonstration requires to make an assumption or guess as to what could $id_A$ be. Specifically, the scheme of the proof is: assume $id_A$ = $1_A$, see that it works with the definition of an identity morphism, show that the identity morphism is unique, and, in conclusion, $id_A$ can only be $1_A$. What I'm looking for, nonetheless, is a somehow more direct proof, that doesn't assume $id_A$ = $1_A$ at the start. I want to place myself in a state of little or no knowledge about sets and functions, and under this assumption, why would I assume $id_A$ = $1_A$ at first? Why not try with $id_\mathbb{Z}$ = $f \colon \mathbb{Z} \to \mathbb{Z}, x \mapsto x^2 + 1$, for example? It wouldn't work, but I don't have any reason to think that $1_\mathbb{Z}$ is a better guess for $id_\mathbb{Z}$.  I suppose that the proof for which I'm asking would work for categories of sets with additional structure, and probably for posets as well, although I'm not clear as to what modifications it would require to work.  Thanks. Is there a list of error codes for Minecraft? Similar to  on the bugs in Minecraft, I would like a complete list of what error codes exist in Minecraft and how they occur, as well as how they can be resolved. Multi document printing from document library? Is there any one have solution for print multiple documents from Document library? User has option to select multiple documents or single document or inside folder.  I found commercial product "Unbounded Printing Services for SharePoint". But is so expensive. Please share me if you have any thoughts. The Data Interoperability Tools .tbx appears in my system toolboxes, however when I double click quick export or import, a red X appears.  I have gone into the administrator settings and the tool doesn't appear to be installed. How do I install it?  I am using ArcGIS 10.2 for desktop trial.  After I attempt to evaluate extensions, I receive a message which states all requested features have already been authorized on this machine. 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My attempt:  We only have to prove this assertion for a non-negative Lebesgue measurable function $f$ because the result will follow for all Lebesgue measurable functions.  Furthermore, since any Lebesgue measurable function can be approximated by a sequence of non-negative, monotonically increasing, Lebesgue measurable simple functions $s_{n}$, we only need to prove the claim for an arbitrary Lebesgue measurable simple function.  So, let $s: \mathbb{R} \to [-\infty, \infty] $ be a Lebesgue measurable simple function.  We can write $s$ canonically as $$ s(x) = \sum \limits_{i = 1}^{n} \alpha_{i} \chi_{A_{i}}(x)$$ where $\alpha_{i} \in [-\infty, \infty]$, $\bigcup \limits_{i = 1}^{n} A_{i} = \mathbb{R}$, and $A_{i} \cap A_{j} = \emptyset$ if $i \neq j$.  For each $i$, since $A_{i}$ is Lebesgue measurable, we can find a Borel set $B_{i}$ such that $A_{i} \subseteq B_{i}$ and $m(B_{i} \setminus A_{i}) = 0$.  Clearly, this implies that $\bigcup \limits_{i = 1}^{n} B_{i} = \mathbb{R}$.  Now we just need the $B_{i}$'s to be pairwise disjoint, with each $B_{i}$ still retaining $A_{i}$.  To make them pairwise disjoint, I constructed the following sets:  $\tilde{B_{i}} = [B_{i} \setminus (\bigcup \limits_{j \neq i} B_{j})] \cup A_{i}$.  This construction gives us that the $\tilde{B_{i}}$'s are pairwise disjoint (I think....) and $A_{i} \subseteq \tilde{B_{i}}$.  But I don't know that $\tilde{B_{i}}$ is necessarily still a Borel set. :( :(  Am I approaching this problem all wrong? Topological/Geometrical justification for $\text{CFT}_2$ being special How do you access the Shrine of Winter? In Dark Souls 2 there is a 'Shrine of Winter' near Sinners Rise. When you approach the shrine it will not let you open the door and a message pops up that says      Seek Mightier Souls   What does this mean and how do you open this shrine?