Question

Let the table $$ \begin{array}{c|cccccc} x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline \text { Frequency } & 6 & 10 & 14 & 13 & 6 & 1 \end{array} $$ represent a summary of a sample of size 50 from a binomial distribution having \(n=5\). Find the mle of \(P(X \geq 3)\). For the data in the table, using the \(\mathrm{R}\) function pbinom determine the realization of the mle.

Short answer

The mle of \(P(X \geq 3)\) is found by computing the likelihood function, finding its maximum, and then using R function pbinom() to determine the realization of the mle. The exact value can't be found without the actual computation in R with the given dataset.

Step-by-step solution

Compute the Likelihood Function

The likelihood function \(L(\theta)\) is given by \(L(\theta) = \prod_{k=0}^{n} f_k(\theta)^{y_k}\), where \(f_k(\theta)\) is the binomial distribution probability function, and \(y_k\) is the frequency of occurrence for each value of \(k\). We have \(f_k(\theta) = C(n, k) \theta^{k}(1-\theta)^{n-k}\) and \(y_k\) as the given frequencies for each \(k\). Let's now plug these values into the likelihood function formula and simplify.

Find the Maximum Likelihood Estimate

The maximum likelihood estimate for \(\theta\) is obtained by taking the derivative of the log-likelihood function \(l(\theta) = \log L(\theta)\) and setting it equal to zero. Then solve for \(\theta\). This is a more practical method as the log function transforms the product into a sum, making calculations easier.

Use the R function pbinom

To find a realization of the maximum likelihood estimate in R, you have to use the function pbinom(). This function provides the cumulative distribution function for a binomial random variable.

Key concepts

Likelihood Function

The concept of a likelihood function is fundamental in the field of statistics, particularly when we wish to estimate parameters of a model that best fit our data. In the context of a binomial distribution, the likelihood function, denoted as \(L(\theta)\), represents the probability of observing the given data as a function of the probability parameter \(\theta\).To construct the likelihood function for a binomial distribution, you consider the probability of achieving each observed outcome under varying assumptions of the parameter \(\theta\). Specifically, if you are given the number of successes in a fixed number of trials (with each trial resulting in a success or failure), the likelihood function is the product of the probabilities for each of those outcomes, each raised to the power of its observed frequency. The likelihood function for a binomial distribution with parameters \(n\), number of trials, and \(p\), the probability of success, is given by:\[ L(p; x_1, x_2, \.\., x_k) = prod_{i=1}^{k} binom{n}{x_i} p^{x_i}(1-p)^{n-x_i} \]where \(x_i\) represents the number of successes in the \(i\)th trial.The maximum likelihood estimation (MLE) is a method used to find the value of parameter \(p\) that maximizes this likelihood function, thus making the observed data 'most likely'. The MLE for the binomial distribution is found by taking the derivative of the log-likelihood function (logarithm of the likelihood function), setting it to zero, and solving for \(p\). This is typically easier as the log transformation turns the product into a sum, simplifying the differentiation process.

The ultimate goal is to find the parameter value that gives the highest likelihood of observing the actual data, embodying the concept of 'fitting' the model to the observed outcomes.

Binomial Distribution

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent experiments, each asking a yes/no question, and each with its own boolean-valued outcome: success/failure, win/lose, on/off, etc. It forms the basis of the binomial test of statistical significance and is one of the most fundamental distributions in statistics.A binomial distribution is fully characterized by two parameters: the number of trials, \(n\), and the probability of success in a single trial, \(p\). The probability mass function (pmf) for a binomial distribution, that is, the probability of getting exactly \(k\) successes in \(n\) trials is given by:\[ P(X = k) = binom{n}{k} p^k(1-p)^{n-k} \]where \(X\) is a random variable representing the number of successes, and \(binom{n}{k}\) is the binomial coefficient, which calculates the number of ways to choose \(k\) successes from \(n\) trials.In the exercise provided, we're considering the multiple frequencies of different numbers of successes in a total of 50 trials of a binomial experiment with \(n=5\). To find the maximum likelihood estimate (MLE) for \(P(X \geq 3)\), you must consider the frequencies for 3, 4, and 5 successes in your calculation while applying the principles of MLE.

R Statistical Software

R statistical software is an open-source programming language and environment widely used for statistical computing and graphics. It is highly extensible, offering a variety of statistical and graphical techniques, including linear and nonlinear modeling, classical statistical tests, time-series analysis, classification, clustering, and more. It has become a staple tool for statisticians for data analysis and developing statistical software.One of the reasons for R's popularity in statistics is its vast library of packages that serve various statistical functions. In the case of our exercise, the 'pbinom' function is crucial. This function in R calculates the cumulative probability distribution function (CDF) for a binomial distribution, which gives the probability of having a number of successes less than or equal to a particular value \(k\).To utilize the 'pbinom' function effectively, one needs to specify the number of successes, the number of trials, and the probability of success. For the calculation of the MLE, this is especially useful as it automates the computation process for the binomial probabilities over a range of values, making it easier to find the maximum likelihood estimate. Long calculations done manually can introduce errors, but using R ensures consistency and accuracy in the statistical analysis.

Assisting with homework and academic exercises, R's functions, like 'pbinom,' not only offer students a powerful computational tool but also help them understand the underlying statistical concepts and how they can be applied in practical settings.