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 \(P(X \geq 3)\)

Short answer

The MLE of \(P(X \geq 3)\) is 0.4.

Step-by-step solution

Identify Frequencies

Read the data from the given frequency table. Notice that the 'x' row represents the possible outcomes while the 'Frequency' row represents the number of times these outcomes occurred in the sample. It is given that \(n=5\). We are interested in \(P(X \geq 3)\), hence, we need to use the frequencies for x = 3, 4 and 5.

Calculate Relative Frequencies

Calculate the relative frequencies for x = 3, 4, and 5 by dividing the frequency of each by the total size of the sample, which is 50. The relative frequency for x = 3 is \(\frac{13}{50}\), for x = 4 it's \(\frac{6}{50}\), and for x = 5 it's \(\frac{1}{50}\).

Sum Relative Frequencies

Sum up the relative frequencies calculated in the previous step. This gives us the estimated probability \(\hat{p}\) as the MLE of \(P(X \geq 3)\). So, \(\hat{p} = \frac{13}{50} + \frac{6}{50} + \frac{1}{50} = \frac{20}{50} = 0.4\).

Key concepts

Binomial Distribution

The binomial distribution is a fundamental probability distribution used in statistics and probabilistic studies. It models the number of successes in a fixed number of independent Bernoulli trials, which are experiments that have exactly two outcomes: success and failure. In our example, a sample size of 50 was taken from a binomial distribution where the number of trials, denoted by \( n \), is 5.

• The number of trials (\( n \)) is the number of times an experiment is performed, which is 5 for each observation in the given exercise.
• Each trial has exactly two possible outcomes: success ("success" here is achieving a count of outcomes with \( x \geq 3\)) or failure ("failure" would be outcomes where \( x < 3 \)).
• The probability of success (\( p \)) remains constant for each trial.

When working with the binomial distribution, we often use it to determine the probability of obtaining a certain number of successes across multiple trials. In this problem, we're particularly interested in calculating the probability for cases where the number of successes is 3, 4, or 5.

Relative Frequency

The concept of relative frequency helps in understanding how frequently a particular outcome occurs, relative to the total number of trials. In this exercise, the relative frequency is obtained by dividing the frequency of each outcome by the total number of observations, which is given as 50. This provides a proportion that is used to estimate probabilities.

• For \( x = 3 \), the relative frequency is calculated as \( \frac{13}{50} \).
• For \( x = 4 \), it's \( \frac{6}{50} \).
• For \( x = 5 \), the calculation is \( \frac{1}{50} \).

These relative frequencies are essential for estimating probabilities in practical scenarios. They reflect the observed data's behavior and help understand how common or rare certain outcomes are relative to the entire dataset. The total of relative frequencies for \( x \ge 3 \) will give us an important probability value, which is used in the estimation process.

Probability Estimation

Probability estimation involves predicting the likelihood of an event based on sample data. In the exercise, the objective was to estimate the probability \( P(X \geq 3) \) using relative frequencies obtained from the data. This is achieved by summing the relative frequencies of outcomes where \( x = 3, 4, \) and \( 5 \).

• Calculate individual relative frequencies for each desired outcome: \( \frac{13}{50}, \frac{6}{50}, \frac{1}{50} \).
• Add these relative frequencies to estimate the probability: \( \hat{p} = \frac{13}{50} + \frac{6}{50} + \frac{1}{50} = \frac{20}{50} = 0.4 \).

The MLE (Maximum Likelihood Estimation) approach is used here, which seeks to find the probability value that would make the observed data most probable. Therefore, the estimated probability of getting \( X \geq 3 \) in a trial is 0.4. This method of probability estimation is commonly employed when working with sample data, as it provides a relatively simple and intuitive approach to predicting the likelihood of events based on observed results.