mle of bernoulli

which, except for the factor \(\dfrac{n!}{x!(n-x)! Bernoulli Distribution : In this notation, We calculate the probability given x and theta. If the outcome is X = 3, the likelihood is, \(\begin{align} L(p;x) &= \dfrac{n!}{x!(n-x)!} The score function for the Bernoulli log-likelihood is S(θ|x)= ∂lnL(θ|x) ∂θ = 1 θ Xn i=1 xi− 1 1−θ à n− Xn i=1 xi! likelihood : A probability of happening possibility of an event. Gregory Gundersen is a PhD candidate at Princeton. Suppose that \(X = (X_1, X_2, \dots, X_n)\) are iid observations from a Poisson distribution with unknown parameter \(\lambda\). Fisher information. We often call \(\hat{p}\) the sample proportion to distinguish it from p, the “true” or “population” proportion. Maximum likelihood estimation is a popular method for estimating parameters in a statistical model. 3.2 MLE: Maximum Likelihood Estimator Assume that our random sample X 1; ;X n˘F, where F= F is a distribution depending on a parameter . ); Minimize the negative log-likelihood èMLE parameter estimation i.e. Because the natural log is an increasing function, maximizing the loglikelihood is the same as maximizing the likelihood. For example, suppose that \(X_1, X_2, . To maximize \(L(\theta ; x)\) with respect to \(\theta\): These computations can often be simplified by maximizing the loglikelihood function. Maximum Likelihood Estimation and the E-M Algorithm. First, we … stream MLE tells us which curve has the highest likelihood of fitting our data. endstream L( jx) = f(xj ); 2 : (1) The maximum likelihood estimator (MLE), ^(x) = argmax L( jx): (2) Suppose now that we have a sample of iid binomial random variables. Finding MLE’s usually involves techniques of differential calculus. The method of maximum likelihood was first proposed by the English statistician and population geneticist R. A. Fisher. This asymptotic variance in some sense measures the quality of MLE. << /Filter /FlateDecode /Length 2300 >> This method is known as maximum likelihood estimation or MLE for short. 2.Each trial has only two possible outcomes, \success" and \failure". But collapsing the data in this way may limit our ability to diagnose model failure, i.e. laudantium assumenda nam eaque, excepturi, soluta, perspiciatis cupiditate sapiente, adipisci quaerat odio For instance, if F is a Normal distribution, then = ( ;˙2), the mean and the variance; if F is an Exponential distribution, then = , the rate; if F is a Bernoulli … Assumptions. The goal of MLE is to infer ... First, each coin flipping follows a Bernoulli distribution, so the likelihood can be written as: In the formula, xi means a single trail (0 or 1) and x means the total number of heads. For example, if is a parameter for the variance and ^ is the maximum likelihood estimator, then p ^ is the maximum likelihood estimator for the standard deviation. solve the resulting equation for \(\theta\). Since data is usually samples, not counts, we will use the Bernoulli rather than the binomial. And thus a Bernoulli distribution will help you understand MLE for logistic regression. In the second one, $\theta$ is a continuous-valued parameter, such as the ones in Example 8.8. This asymptotic variance in some sense measures the quality of MLE. Bernoulli trials are one of the simplest experimential setups: there are a number of iterations of some activity, where each iteration (or trial) may turn out to be a "success" or a "failure". We do this in such a way to maximize an associated joint probability density function or probability mass function.. We will see this in more detail in what follows. Hence,thesampleaverageistheMLEforθin the Bernoulli model. to show that ≥ n(ϕˆ− ϕ 0) 2 d N(0,π2) for some π MLE MLE and compute π2 MLE. The fact that the MLE based on n independent Bernoulli random variables and the MLE based on a single binomial random variable are the same is not surprising, since the binomial is the result of n independent Bernoulli trials anyway. Maximum likelihood estimation (MLE) is a method to estimate the parameters of a random population given a sample. The maximum likelihood estimator determined the asymptotic properties and is especially good in the large-sample situation. where k is a constant that does not involve the parameter p. In the future, we will omit the constant, because it's statistically irrelevant. In this example we use the CmdStan example model bernoulli.stan and data file bernoulli.data.json. For repeated Bernoulli trials, the MLE \(\hat{p}\) is the sample proportion of successes. Now, let's check the maximum likelihood estimator of \(\sigma^2\). That is, yi = 1 with probability and zero with probability 1-0. a) Write down the log likelihood function for the sample and show that the MLE is the sample mean. paper,the maximum likelihood andBayesian methodsare usedfor estimating parameter ofBernoulli distribution, i.e. We assume to observe inependent draws from a Poisson distribution. Definition 1. Maximum Likelihood Estimation and the E-M Algorithm. 5 We will denote the value of \(\theta\) that maximizes the likelihood function by \(\hat{\theta}\), read “theta hat.”\(\hat{\theta}\) is called the maximum-likelihood estimate (MLE) of \(\theta\). First, note that we can rewrite the formula for the MLE as: Maximum Likelihood Estimation. For example, if is a parameter for the variance and ˆ is the maximum likelihood estimate for the variance, then p ˆ is the maximum likelihood estimate for the standard deviation. Bernoulli MLE Estimation For our first example, we are going to use MLE to estimate the p parameter of a Bernoulli distribution. Bernoulli MLE Estimation For our first example, we are going to use MLE to estimate the p parameter of a Bernoulli distribution. Bernoulli trials are one of the simplest experimential setups: there are a number of iterations of some activity, where each iteration (or trial) may turn out to be a "success" or a "failure". Let’s start with Bernoulli distribution !! Most commonly, data follows a Gaussian distribution, which is why I’m dedicating a post to likelihood estimation for Gaussian parameters. Before we can look into MLE, we first need to understand the difference between probability and probability density for continuous variables. Remember that the support of the Poisson distribution is the set of non-negative integer numbers: To keep things simple, we do not show, but we rather assume that the regula… The goal of MLE is to infer ... First, each coin flipping follows a Bernoulli distribution, so the likelihood can be written as: In the formula, xi means a single trail (0 or 1) and x means the total number of heads. Relative Figure 8.1 - The maximum likelihood estimate for $\theta$. In general, whenever we have repeated, independent Bernoulli trials with the same probability of success p for each trial, the MLE will always be the sample proportion of successes. We’ve discussed Maximum Likelihood Estimation as a method for finding the parameters of a distribution in the context of a Bernoulli trial,. statistics define a 2D joint distribution.) In a probit model, the output variable is a Bernoulli random variable (i.e., a discrete variable that can take only two values, either or ). We do this in such a way to maximize an associated joint probability density function or probability mass function.. We will see this in more detail in what follows. to show that ≥ n(ϕˆ− ϕ 0) 2 d N(0,π2) for some π MLE MLE and compute π2 MLE. From the data on T trials, we want to estimate the probability of "success". In each sample, we have n = 100 draws from a Bernoulli distribution with true parameter p 0 = 0.4. Lorem ipsum dolor sit amet, consectetur adipisicing elit. Introduction. Example: estamate MLE for model bernoulli.stan by optimization¶. Answer and Explanation: Become a Study.com member to unlock this answer! Bernoulli MLE Estimation For our first example, we are going to use MLE to estimate the p parameter of a Bernoulli distribution. Adding the binomial random variables together produces no loss of information about p if the model is true. ignoring the constant terms that do not depend on \(\lambda\), one can show that the maximum is achieved at \(\hat{\lambda}=\sum\limits^n_{i=1}x_i/n\). Now, let's check the maximum likelihood estimator of \(\sigma^2\). x��]�ܶ��~���E-�_���n�Ɓ��M�A��=�֊I����b8�VZ��(�>�����p������͸��*��g�*���BRQd7��7�9��3�f�Ru�� ���`�y?�C5��n~���qj�B 6Ψ0*˥����֝����5�v��׮��o��:x@��ڒg�0�X��^W'�yKm)J��s�iaU�+N��x�ÈÃu��| ��J㪮u��C��V�����7� {׹v@�����n#'�A������U�.p��:_�6�_�I�4���0ԡw��QW��c4H�IJ�����7���5��iO�[���PW. voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos We compute the MLE separately for each sample and plot a histogram of these 7000 MLEs. . p^x(1-p)^{n-x}\\ &= \dfrac{5!}{3!(5-3)! %PDF-1.5 The Since \(X_1, X_2, \dots, X_n\) are iid random variables, the joint distribution is, \(L(p;x)\approx f(x;p)=\prod\limits_{i=1}^n f(x_i;p)=\prod\limits_{i=1}^n p^x(1-p)^{1-x}\). Since \(\sum\limits_{i=1}^n x_i\) is the total number of successes observed in the n trials, \(\hat{p}\) is the observed proportion of successes in the n trials. 17 0 obj \cdots x_{n} !} What is the MLE of the probability of success θ is it is know that θ is at most 1/4 Homework Equations f(x,θ) = θ x (1-θ) 1-x The Attempt at a Solution Now, I know how to find the likelihood and use it to solve for the MLE. first calculate the derivative of \(L(\theta ; x)\) with respect to \(\theta\). . ML for Binomial Section Suppose that X is an observation from a binomial distribution, X ∼ Bin( n , p ), where n is known and p is to be estimated. 2.2 Estimation of the Fisher Information If is unknown, then so is I X( ). The Bernoulli distribution models events with two possible outcomes: either success or failure. endobj As we know from statistics, the specific shape and location of our Gaussian distribution come from σ and μ respectively. In more formal terms, we observe the first terms of an IID sequence of Poisson random variables. This example suggests that it may be reasonable to estimate an unknown parameter \(\theta\) by the value for which the likelihood function \(L(\theta ; x)\) is largest. You get the same value by maximizing the binomial loglikelihood function, \(l(p;x)=k+x\text{ log }p+(n-x)\text{ log }(1-p)\). In frequentist inference, MLE is a special case of an extremum estimator, with the objective function being the likelihood. Asymptotic Normality of Maximum Likelihood Estimators Under certain regularity conditions, maximum likelihood estimators are "asymptotically efficient", meaning that they achieve the Cramér–Rao lower bound in the limit. We want to show the asymptotic normality of MLE, i.e. If the probability of Success event is P then the probability of Failure would be (1-P). 1.The experiment is repeated a xed number of times (n times). For a Bernoulli distribution, d/(dtheta)[(N; Np)theta^(Np)(1-theta)^(Nq)]=Np(1-theta)-thetaNq=0, (1) so maximum likelihood occurs for theta=p. x�c```b``������#� � `620�3�YΕ+����7M&��*4AH�YP'7��, � 2ll?�r�����]�Bl��y](qy�Q� ��� 2 Outline MLE: Maximum Likelihood Estimators EM: the Expectation Maximization Algorithm Relative Entropy. In each sample, we have \(n=100\) draws from a Bernoulli distribution with true parameter \(p_0=0.4\). �ɅT�?���?��, ��V����෸68L�E*RG�H5S8HɊHD���J֌���4�-�>��V�'�Iu6ܷ/�ȸ�R��"aY.5�"�� ���3\�,�����!�a�� 3���� V 8:��%���Z�+�4o��ڰ۸�MQ����� ���j��sR��B)�_-�T���J���#|L���X�J��]Lds�j;���a|Y��M^2#��̶��( The Binary Logistic Regression problem is also a Bernoulli distribution. 16 0 obj s0_�q�,�"Q�F1'"�Q�m8��w�~�;#[�vN��6]�S�s]?T������+]غ�W���Q�UZ�s�����ggfKg�{%�R�k6a���ʢ=��C�͆��߷��_P[��l�sY�@� �2��V:#�C�vI�}7 stream p^x(1-p)^{n-x}\). We assume to observe inependent draws from a Poisson distribution. Two estimates I^ of the Fisher information I X( ) are I^ 1 = I X( ^); I^ 2 = @2 @ 2 logf(X j )j =^ where ^ is the MLE of based on the data X. I^ 1 … 3.2 MLE: Maximum Likelihood Estimator Assume that our random sample X 1; ;X n˘F, where F= F is a distribution depending on a parameter . Using large sample sizes (modify n as necessary ) verify, using the Monte Carlo method, the convergence properties of the MLE estimators of the Cauchy distribution (analyze each estimate separately) : In probability: see if estimates seem to convergence to some constant (which one? We want to show the asymptotic normality of MLE, i.e. voluptates consectetur nulla eveniet iure vitae quibusdam? N��"C-B&Wp����s�;��&WF$ Hf�$�ķ�����$� endstream Then take a log for the likelihood: If you line these up on a number line, you can see that : MLE is most accurate if the population parameter is greater than (0.7333 + 0.75) / … Estimation of parameter of Bernoulli distribution using maximum likelihood approach For example, if is a parameter for the variance and ˆ is the maximum likelihood estimate for the variance, then p ˆ is the maximum likelihood estimate for the standard deviation. , X_{10}\) are an iid sample from a binomial distribution with n = 5 and p unknown. Thus \(X\sim Bin(50,p)\) and the MLE is \(\hat{p}=x/n\), the observed proportion of successes across all 50 trials. , which isdefined asthe probability of success event for two possible outcomes.The maximum likelihood and Bayesian estimators of Bernoulli parameter are derived,for the Bayesian estimator the Beta prior is used. The method of maximum likelihood was first proposed by the English statistician and population geneticist R. A. Fisher. << /Type /XRef /Length 67 /Filter /FlateDecode /DecodeParms << /Columns 4 /Predictor 12 >> /W [ 1 2 1 ] /Index [ 16 48 ] /Info 14 0 R /Root 18 0 R /Size 64 /Prev 96781 /ID [<8a7c60dad2128f758c0ffd96cb0473f8>] >> The likelihood function is, \(L(p;x)=\dfrac{n!}{x!(n-x)!} It was introduced by R. A. Fisher, a great English mathematical statis-tician, in 1912. Main assumptions and notation. Fisher information. Suppose that X is an observation from a binomial distribution, X ∼ Bin(n, p ), where n is known and p is to be estimated. I described what this population means and its relationship to the sample in a previous post. \\ &=\dfrac{\lambda^{\sum_{i=1}^{n} x_{i}} e^{-n \lambda}}{x_{1} ! ?�.� 2�;�U��=�\��]{ql��1&�D���I|@8�O�� ��pF��F܊�'d��K��`����nM�{?���D�3�N\�d�K)#v v�C ��H Ft������\B��3Q�g�� Therefore, the maximum likelihood estimator of \(\mu\) is unbiased. Since each Xi is actually the total number of successes in 5 independent Bernoulli trials, and since the Xi’s are independent of one another, their sum \(X=\sum\limits^{10}_{i=1} X_i\) is actually the total number of successes in 50 independent Bernoulli trials. where the constant at the beginning is ignored. What are parameter estimates? Conditional on a vector of inputs , we have that where is the cumulative distribution function of the … 18 0 obj 3.The probability of success remains the same for each trial. For repeated Bernoulli trials, the MLE \(\hat{p}\) is the sample proportion of successes. the MLE ; the median of the sample Using large sample sizes (modify n as necessary ) verify, using the Monte Carlo method, the convergence properties of the MLE estimators of the Cauchy distribution (analyze each estimate separately) : It was introduced by R. A. Fisher, a great English mathematical statis-tician, in 1912. endobj Differentiating the log of L(p ; x) with respect to p and setting the derivative to zero shows that this function achieves a maximum at \(\hat{p}=\sum\limits_{i=1}^n x_i/n\). This is true regardless of whether we know the outcomes of the individual trials \(X_1, X_2, \dots , X_n\), or just the total number of successes for all trials \(X=\sum\limits^n_{i=1} X_i\). Arcu felis bibendum ut tristique et egestas quis: Suppose that an experiment consists of n = 5 independent Bernoulli trials, each having probability of success p. Let X be the total number of successes in the trials, so that \(X\sim Bin(5,p)\). The basic idea behind maximum likelihood estimation is that we determine the values of these unknown parameters. First, we … For instance, if F is a Normal distribution, then = ( ;˙2), the mean and the variance; if F is an Exponential distribution, then = , the rate; if F is a Bernoulli … The likelihood function is the density function regarded as a function of . The likelihood for p based on X is defined as the joint probability distribution of \(X_1, X_2, \dots, X_n\). to check whether the binomial model is really appropriate. Deriving the Maximum Likelihood Estimate for Bernoulli −l(p)=−logn x ⎛ ⎝ ⎜ ⎞ ⎠ ⎟−xlog(p)−(n−x)log(1−p) € dl(p) dp =0− x p − −(n−x) 1−p 0=− x p + n−x 1−p 0= −x(1−p)+p(n−x) p(1−p) 0=−x+px+pn−px 0=−x+pn € p ˆ = x n The proportion of positives! 2 Outline MLE: Maximum Likelihood Estimators EM: the Expectation Maximization Algorithm Relative Entropy. 3 Maximum Likelihood Estimators Learning From Data: MLE. x�cbd�g`b`8 $��A,c �x ��\�@��HH/����z ��H��001��30 �v� The possible outcomes are exactly the same for each trial. For the simple probability models we have seen thus far, however, explicit formulas for MLE’s are available and are given next. }\), is identical to the likelihood from n independent Bernoulli trials with \(x=\sum\limits^n_{i=1} x_i\). The multivariate Bernoulli distribution discussed in Whittaker (1990), which will be studied in Section 1.3, has a probability density function involving terms representing third and higher order moments of the random vari-ables, which are also referred to as clique effects. On top of this histogram, we plot the density of the theoretical asymptotic sampling distribution as a solid line. Let us find the maximum likelihood estimates for the observations of Example 8.8. Suppose that \(X = (X_1, X_2, \dots, X_n)\) represents the outcomes of n independent Bernoulli trials, each with success probability p . We compute the MLE separately for each sample and plot a histogram of these 7000 MLEs. 3��p�@�a���L/�#��0 QL�)��J��0,i�,��C�yG�]5�C��.�/�Zl�vP���!���5�9JA��p�^? The maximum likelihood estimate for a parameter mu is denoted mu^^. The maximum likelihood estimate for a parameter mu is denoted mu^^. This is simply the fraction of examples, for a given class, that contain the particular feature. Then take a log for the likelihood: %���� In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability n) is the MLE, then ^ n˘N ; 1 I Xn ( ) where is the true value. Of course, it is somewhat silly for us to try to make formal inferences about \(\theta\) on the basis of a single Bernoulli trial; usually, multiple trials are available. But since the likelihood function is regarded as a function only of the parameter p, the factor \(\dfrac{n!}{x!(n-x)! If our experiment is a single Bernoulli trial and we observe X = 1 (success) then the likelihood function is \(L(p ; x) = p\). Whenever we have independent binomial random variables with a common p , we can always add them together to get a single binomial random variable. ��-�� Case Study: The Ice Cream Study at Penn State, Understanding Polytomous Logistic Regression, Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris, Duis aute irure dolor in reprehenderit in voluptate, Excepteur sint occaecat cupidatat non proident. where “log” means natural log (logarithm to the base e). Then, the principle of maximum likelihood yields a choice of the estimator ^ as the value for the parameter that makes the observed data most probable. \end{aligned}, By differentiating the log of this function with respect to \(\lambda\), that is by differentiating the Poisson loglikelihood function, \(l(\lambda;x)=\sum\limits^n_{i=1}x_i \text{ log }\lambda-n\lambda\). statistics probability-distributions maximum-likelihood log-likelihood  Share. The main elements of a maximum likelihood estimation problem are the following: 1. a sample , that we use to make statements about the probability distribution that generated the sample; 2. the sample is regarded as the realization of a random vector , whose distribution is unknown and needs to be estimated; 3. there is a set of real vectors (called the parameter space) whose elements (called parameters) are put into correspondence … endobj }p^3 (1-p)^{5-3}\\ & \propto p^3(1-p)^2\\ \end{align}\). �0���. The loglikelihood often has a much simpler form than the likelihood and is usually easier to differentiate. If ˆ(x) is a maximum likelihood estimate for , then g( ˆ(x)) is a maximum likelihood estimate for g( ). Since data is usually samples, not counts, we will use the Bernoulli rather than the binomial. Non-technical question about maximum likelihood estimation / … 2.1 Maximum likelihood parameter estimation In this section, we discuss one popular approach to estimating the parameters of a probability density function. �"ۺ:bRQx7�[uipRI������>t��IG�+?�8�N��h� ��wVD;{heջoj㳶��\�:�%~�%��~y�6�mI� ����-Èo�4�ε[���j�9�~H���v.��j[�� ���+�߅�����1`&X���,q ��+� Asymptotic normality of MLE. What is the Maximum Likelihood Estimation. Thus, for a Poisson sample, the MLE for \(\lambda\) is just the sample mean. Serously? Maximum Likelihood Estimation (MLE) example: Bernouilli Distribution Link to other examples: Exponential and geometric distributions Observations : k successes in n Bernoulli trials. Thus, the probability mass function of a term of the sequence is where is the support of the distribution and is the parameter of interest (for which we want to derive the MLE). no explicit formulas for MLE’s are available, and we will have to rely on computer packages to calculate the MLE’s for us. A graph of \(L(p;x)=p^3(1-p)^2\) over the unit interval \(p ∈ (0, 1)\) looks like this: It’s interesting that this function reaches its maximum value at \(p = .6\). Thus the MLE is again \(\hat{p}=x/n\), the sample proportion of successes. If ˆ(x) is a maximum likelihood estimate for , then g( ˆ(x)) is a maximum likelihood estimate for g( ). endobj This function reaches its maximum at \(\hat{p}=1\). The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. 20 0 obj The basic idea behind maximum likelihood estimation is that we determine the values of these unknown parameters. Except where otherwise noted, content on this site is licensed under a CC BY-NC 4.0 license. In more formal terms, we observe the first terms of an IID sequence of Poisson random variables. endobj Since a Bernoulli is a discrete distribution, the likelihood is the probability mass function. Asymptotic normality of MLE. Try it for yourself. In most of the probability models that we will use later in the course (logistic regression, loglinear models, etc.) 1) Consider an independent sample of size N drawn from the Bernoulli distribution, with probability parameter 0. This is where estimating, or inf e rring, parameter comes in. Step one of MLE is to write the likelihood of a Bernoulli as a function that we can maximize. Bernoulli Experiment with n Trials Here are the rules for a Bernoulli experiment. Maximum Likelihood Estimation INFO-2301: Quantitative Reasoning 2 Michael Paul and Jordan Boyd-Graber MARCH 7, 2017 INFO-2301: Quantitative Reasoning 2 j Paul and Boyd-Graber Maximum Likelihood Estimation j 1 of 9 The maximum likelihood estimator (MLE), ^(x) = argmax L( jx): (2) Note that if ^(x) is a maximum likelihood estimator for , then g(^ (x)) is a maximum likelihood estimator for g( ). Excepturi aliquam in iure, repellat, fugiat illum The MLE satisfies S(ˆθ mle|x)=0,which after a little algebra, produces the MLE ˆθ mle= 1 n Xn i=1 xi. a dignissimos. In both cases, the maximum likelihood estimate of $\theta$ is the value that maximizes the likelihood function. Therefore, the maximum likelihood estimator of \(\mu\) is unbiased. statistics define a 2D joint distribution.) A couple of things to know about this study ... How complex can the models get? In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a probability distribution by maximizing a likelihood function, so that under the assumed statistical model the observed data is most probable. 2018, Aug 26 . This approach is called maximum-likelihood (ML) estimation. The use of maximum likelihood estimation to estimate the parameter of a Bernoulli random variable. Note that in some textbooks the authors may use π instead of p. For repeated Bernoulli trials, the MLE \(\hat{p}\) is the sample proportion of successes. << /Linearized 1 /L 97144 /H [ 922 192 ] /O 20 /E 61819 /N 6 /T 96780 >> Odit molestiae mollitia If we observe X = 0 (failure) then the likelihood is \(L(p ; x) = 1 − p\), which reaches its maximum at \(\hat{p}=0\). 19 0 obj Two independent bernoulli trials resulted in one failure and one success. 2.1 Maximum likelihood parameter estimation In this section, we discuss one popular approach to estimating the parameters of a probability density function. Your data sample gives a MLE estimate of 0.75. First, note that we can rewrite the formula for the MLE as: Maximum Likelihood Estimation. Maximum Likelihood Estimation Lecturer: Songfeng Zheng 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for an un-known parameter µ. You have done two different Bayesian calcs, call them B1 and B2, producing estimates of 0.7333 and 0.6. The Bernoulli distribution is a special case of the binomial distribution with = [3] The kurtosis goes to infinity for high and low values of p , {\displaystyle p,} but for p = 1 / 2 {\displaystyle p=1/2} the two-point distributions including the Bernoulli distribution have a lower excess kurtosis than any other probability distribution, namely −2. x_{2} ! stream << /Contents 21 0 R /MediaBox [ 0 0 612 792 ] /Parent 36 0 R /Resources 29 0 R /Type /Page >> The maximum likelihood method finds that estimate of a parameter which maximizes the probability of observing the data given a specific model for the data. the MLE ; the median of the sample.

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