Suppose $X1$ and $X2$ are iid Bernoulli random variable with parameter $p$ where it is known that $1/3&lt;=p&lt;=2/3$. Find the maximum likelihood estimator $p*$ of $p$ based on $X1$ and $X2$. I could find the MLE and it doesn't include the fact anywhere that the parameter $p$ lies in a certain range. How do I proceed?
123437
Suppose $X_1, X_2, \dots, X_n$ are iid Bernoulli(p) random variables. How do you find the restricted maximum likelihood for p where $0&lt;p&lt;0.5$?  My work so far:  Write out the likelihood: $L=p^{\sum x} (1-p)^{n-\sum x}$  Take the log: $\ln(L) = \sum x\ln(p)+(n-\sum x)\ln(1-p)$  Take the derivative and set equal to zero: $\frac{\sum x-np}{p(1-p)}=0$.  Now I'm stuck. I don't know how to incorporate the restrictions for $p$.
39573
How can I see that the maximum likelihood approach finds the parameter values of the probability distribution that maximize the probability of the observed sample? Maximum likelihood is not the maximum of probability in general because for, say, two continuous random variables $X_1, X_2$ we have $P(X_1=x_1,X_2=x_2)=0$.
85974
Suppose that $T_1,…,T_{10}$ are iid $Exp(λ)$ and the goal is to test if $H_0:λ≤1$ versus $H_a:λ≥2$. Suppose that the test statistic is $S=\sum_{i=1}^{10}T_i$, and rejection of the null occurs when $S≤7$.  I realize that $T_1...T_{10}$ is a gamma distribution  of $S =Gamma(10, \lambda$), but I am not sure how to use this information to find the p-value for the type 1 error.
45628
Suppose that $X_1,X_2,...,X_n$ is a random sample from a normal distribution, $X_i\sim N(\mu,9)$.  Find the UMVUE (uniformly minimum variance unbiased estimator) of $P(X\le c)$ where $c$ is a known constant.  Do this by finding the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$ and apply the Rao-Blackwell theorem with $T=u(X_1)$, where $u(x_1)=1$ if $x\le c$ and zero otherwise.  So clearly the parameter that we're trying to estimate given the problem description can be written as $\Phi\left( \frac{c-\mu}{3} \right)$ where $\Phi$ represents the CDF of the standard normal distribution.  However, I get stuck when I need to try and find the conditional distribution.  Without knowing the joint distribution or the other conditional distribution, how can I possibly find the conditional distribution of $X_1$ given $\bar{X}={\bar{x}}$?
99912
I have 10 iid r.v. with Bernoulli distribution with $X_{i} = 1$ for a positive result. I'm given $\sum_{i=1}^{10} X_i= 1$ and need to find a two-sided 99% confidence interval for $\theta$.   So $\alpha = 0.01$ and $z_{\alpha/2} = 2.575$ based on the standard normal distribution. The variance is given $\sigma^{2} = \hat{\theta}(1-\hat{\theta})$, with $\theta = 1/10$ from the given information. The confidence interval is: $$ \left(\hat{\theta} - z_{\alpha/2}\frac{\sigma}{\sqrt{n}}, \hat{\theta} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\right) $$ so my confidence interval is $(-0.143, 0.343)$. Have I messed up a calculation somewhere? I'm not sure what this confidence interval tells me ... perhaps that the sample size is too small to really get any useful information on $\theta$?
109428
Suppose I want to generate $X_1, \ldots, X_n$ Bernoulli random variables such that:  $$ Corr(X_i, X_j) = \rho $$  for all $i \neq j$.  I am wondering what method I might be able to use (I will be trying to do this in R). I am thinking of using Couplas, but could anyone guide me how to implement this? Thanks.
90558
The full question is,     If X is a Geometric random variable with parameter p, what value of k ∈ {1, 2, . . .}   maximizes P(X = k) = (1 − p)^(k-1) * p? This is referred to as the mode of the distribution.   Through trial and error (plugging in random values), I found that p = 1/2 yields the highest value of P(X=k).  Is there a way to figure this out without trial and error?
122046
What are the maximum-likelihood estimators for the parameters of Student's t-distribution? Do they exist in closed form? A quick Google search didn't give me any results.  Today I am interested in the univariate case, but probably I will have to extend the model to multiple dimensions.  EDIT: I am actually mostly interested in the location and scale parameters. For now I can assume that the degrees of freedom parameter is fixed, and possibly use some numeric scheme to find the optimal value later.
15203
I have a random sample of Bernoulli random variables $X_1 ... X_N$, where $X_i$ are i.i.d. r.v. and $P(X_i = 1) = p$, and $p$ is an unknown parameter.  Obviously, one can find an estimate for $p$: $\hat{p}:=(X_1+\dots+X_N)/N$.  My question is how can I build a confidence interval for $p$?
60841
I am looking for the maximum likelihood estimator (MLE) for the .  I understand the derivation of the MLE for a  and a , but I am unable to derive the MLE equation for the Poisson-binomial distribution.
21440
$X_1, X_2 \dots X_n$ represent independently observed Bernoulli random variables.  $Z_1, Z_2, \dots Z_n$ are unobserved.  $Z_i | \theta_i \sim N(\theta_i,1)$  $\theta_i \sim N(\epsilon, \sigma^2)$  $X_i = \begin{cases} 0 &amp;  Z_i \leq u \\ 1 &amp; Z_i &gt; u \end{cases}$ where $u$ is known.  Show that the EM sequence for estimation of $\epsilon$ is given by:  $\epsilon^{t+1} = \frac{1}{n} \sum_{i=1}^n E[Z_i|X_i, \epsilon^t, \sigma^2]$  $\Phi$ represents CDF of standard normal.  Attempt  $Z_i \sim N(\epsilon, \sigma^2+1)$   \begin{align*} f(X,Z| \epsilon, \sigma^2) &amp;= \prod_{i=1}^n p_i^{x_i} (1-p_i)^{1-x_i}\\ p_i &amp;= P(Z_i \leq u)\\ &amp;=\Phi(\frac{u-\epsilon}{\sqrt{\sigma^2+1}}) \end{align*}  Complete log likeihood = $\sum_{i=1}^nx_i \log p_i + (n-\sum_i x_i) \log(1-p_i)$  I am not able to proceed further.
62578
Let $Z_1, Z_2,... Z_n \sim Ber(\theta)$  Let $X$ be the number of trials before the 1st success. So $X \sim NB(1,\theta)$  First, I was asked to find an MLE of $\theta$ based on $X$. I did this by going through the MLE procedure for a single rv $X$ which yielded my estimator $\frac{1}{x}$.  Next, I was asked to find the MLE for $f(\theta)=\frac{\theta}{1-\theta}$. For this, I used the invariance property of MLEs so since $\frac{1}{x}$ estimates $\theta$, the estimator for $f(\theta)=\frac{1}{x-1}$. My problem with this estimator is that it is undefined when $x=1$.  Where did I go wrong? Thanks.
8183
Suppose we have i.i.d. $n$ observations $(X_1,X_2,...X_n)$ from a   population with density $$f_\theta(x)=\begin{cases}\theta x^{\theta-1}&amp;\text{ if }0\leq x\leq 1\\0&amp;\text{otherwise.}\end{cases}$$ Note that $\theta&gt;0$. If $T_n$ denotes the   maximum likelihood estimator of $\theta$ given that $n$ is the sample   size, then show that $$\text{var}(T_n)\stackrel{n\to\infty}{\longrightarrow} 0$$   This is the problem I am trying to solve: I computed $$T_n=\dfrac{-n}{\sum_{i=1}^{n}\ln(X_i)}$$ But I am stuck in finding $\text{var}(T_n)$ and showing that $\text{var}(T_n)\to0$ as $n\to\infty$.
62728
Let $x_1, x_2... x_n$ be a random sample from a distribution with pdf:  $$f(x;\mu,\sigma)=\frac1{\sigma}\exp\left({-\frac{x-\mu}{\sigma}}\right)\,,-\infty&lt;\mu&lt;\infty;\, \sigma&gt;0;\, x\ge\mu$$  How do I find the MLE for the parameters if both parameters are unknown?  I tried using the usual MLE with likelihood function:  $$L(\mu,\sigma|x_1...x_n)=\frac{1}{\sigma^n}\exp\left({-\frac{\sum{x_i}-n\mu}{\sigma}}\right)$$ But the derivative of this with respect of $\mu$ is a dead end.   I do know that if $\sigma$ is known, the MLE for $\mu$ is $\frac{\sum{x_i}}{n}$ and if $\mu$ is known, the MLE for $\sigma$ is $\frac{\sum{x_i}-n\mu}{n}$.   Do these help? What should be the approach?
31178
$X_1, X_2, . . . , X_n$ are i.i.d. observations from a uniform distribution on the interval $[\theta − \frac{1}{2}, \theta + \frac{1}{2}]$.  Show that any $\theta$ between $X_{max} − \frac{1}{2}$ and $X_{min} + \frac{1}{2}$  maximizes the likelihood, and therefore, can be taken as the MLE.  I am confused...because $f(x|\theta) = \frac{1}{(\theta+\frac{1}{2}) -(\theta -\frac{1}{2})} = 1$ for $\theta-\frac{1}{2} \le x \le\theta+\frac{1}{2}$ and 0 otherwise... Isn't the likelihood function $L(x|\theta) = 1$?  The how can it be maximized between $X_{max} − \frac{1}{2}$ and $X_{min} + \frac{1}{2}$ ?
41990
Let $X_1,X_2,...$ be a sequence of i.i.d Bernoulli random variables   with parameter $p$. Let $N$ be a Poisson random variable with with   parameter $\lambda$ which is independet of the $X_i.$      (a) Find the probability generating function of $Z=\sum_{i=1}^{N}X_i.$      (b) Use (a) to identify the distribution of $Z$.   The pgf of a sum of i.i.d random variables is the product of their respective pgf's so:  $$G_Z(s)=G_{X_1}(s)\cdot G_{X_2}(s)\cdot...\cdot G_{X_N}(s),$$  and sice the $X_{i}$ are Bernoulli distributed with parameter $p$ we have that  $$G_Z(s)=(1-p+ps)^N.$$  I'm not sure how to proceed from here. I can't express my $G_Z(s)$ in terms of a random variable, so what should I do with the $N$?
117149
I am trying to do the following exercise:  Let $\xi_i,\dots, \xi_n$ be independent Bernoulli random variables such that $$ P\{\xi_i=0\}=1-\lambda_i \Delta\\ P\{\xi_i=1\}=\lambda_i \Delta $$ where $\Delta$ is a small number, $\Delta&gt;0,\lambda_i&gt;0$.  Show that $$ P\{\xi_1+\dots+\xi_n=1\}=(\sum_{i=1}^n\lambda_i)\Delta + O(\Delta^2) $$    I already wrote out (as already shown in part of @Marcin Malogrosz answer): $$ P\{\xi_1+\dots+\xi_n) = \sum_i \lambda_i \Delta \prod_{i\neq j} (1-\lambda_j\Delta) \\ \Bigl|\sum_i \lambda_i \Delta \prod_{i\neq j} (1-\lambda_j\Delta) - \sum_i \lambda_i \Delta\Bigr| = \sum_i \lambda_i \Delta \Bigl(1-\prod_{i\neq j} (1-\lambda_j\Delta) \Bigr) $$ But I am stuck here    I am able to show a simplified version with $P\{\xi_i=1\}=\lambda_i\Delta=\lambda\Delta$ (when the $n$ random variables are also identical) as following: $$ P\{\xi_1+\dots+\xi_n=1\}=\sum_i \lambda\Delta (1-\lambda\Delta)^{n-1} $$ using Taylor approximation when $\Delta$ is close to $0$ as $$ \lambda\Delta(1-\lambda\Delta)^{n-1}\Bigr|_{\Delta=0}+\frac{\Delta}{1!}\Bigl\{\lambda(1-\lambda\Delta)^{n-1}-\lambda^2\Delta^2(n-1)(1-\lambda\Delta)^{n-2}\Bigl\}\Bigr|_{\Delta=0} + O(\Delta^2)\\ =\lambda\Delta+O(\Delta^2) $$ and then sum over $i$.    My problem is to show:  $$ \Bigl(1-\prod_{i\neq j} (1-\lambda_j\Delta) \Bigr)=O(\Delta) $$  Thank you in advance