Calculate The Mean For The Discrete Probability Distribution Shown Here

The mean, also known as the expected value, of a discrete probability distribution is a fundamental concept in statistics and probability theory. It represents the average outcome you would expect if you were to repeat the experiment or observation many times. Calculating this mean is crucial for understanding the central tendency of a random variable. This comprehensive guide breaks down the process of calculating the mean for a discrete probability distribution, providing a clear, step-by-step approach with examples and considerations.

Understanding Discrete Probability Distributions

A discrete probability distribution describes the probability of occurrence of each value of a discrete random variable. A discrete random variable is a variable whose value can only take on a finite number of values or a countably infinite number of values. These values are typically integers, but not always.

Key characteristics of a discrete probability distribution:

• Each possible value of the random variable is associated with a probability.
• Each probability must be between 0 and 1, inclusive.
• The sum of all probabilities must equal 1.

Common examples of discrete probability distributions include:

• Bernoulli Distribution: Represents the probability of success or failure in a single trial (e.g., flipping a coin).
• Binomial Distribution: Represents the number of successes in a fixed number of independent trials (e.g., the number of heads in 10 coin flips).
• Poisson Distribution: Represents the number of events occurring in a fixed interval of time or space (e.g., the number of customers arriving at a store in an hour).

Formula for Calculating the Mean of a Discrete Probability Distribution

The mean (μ) of a discrete probability distribution is calculated using the following formula:

μ = Σ [x * P(x)]

Where:

• μ is the mean (expected value).
• x represents each possible value of the random variable.
• P(x) represents the probability of observing the value x.
• Σ represents the summation over all possible values of x.

In simpler terms, you multiply each possible value of the random variable by its corresponding probability and then sum all the results.

Step-by-Step Guide to Calculating the Mean

Here’s a detailed, step-by-step guide to calculating the mean of a discrete probability distribution:

Step 1: Identify the Possible Values of the Random Variable (x)

The first step is to list all the possible values that the random variable can take. These values are usually given in the problem statement or can be determined based on the context of the problem.

Example:

Suppose we are analyzing the number of cars sold by a dealership each day. The possible values for the number of cars sold could be 0, 1, 2, 3, or 4.

Step 2: Determine the Probability of Each Value (P(x))

Next, determine the probability associated with each value of the random variable. These probabilities should be provided or be calculable based on the given information.

Example (Continuing from Step 1):

• P(0) = 0.1 (The probability of selling 0 cars is 0.1)
• P(1) = 0.2 (The probability of selling 1 car is 0.2)
• P(2) = 0.3 (The probability of selling 2 cars is 0.3)
• P(3) = 0.25 (The probability of selling 3 cars is 0.25)
• P(4) = 0.15 (The probability of selling 4 cars is 0.15)

Step 3: Verify That the Sum of Probabilities Equals 1

Before proceeding, ensure that the sum of all probabilities equals 1. This is a fundamental requirement for a valid probability distribution.

Example (Continuing from Step 2):

0. 1 + 0.2 + 0.3 + 0.25 + 0.15 = 1

Since the sum of the probabilities equals 1, we can proceed.

Step 4: Multiply Each Value by Its Probability (x * P(x))

For each value of the random variable, multiply it by its corresponding probability.

Example (Continuing from Step 3):

• 0 * P(0) = 0 * 0.1 = 0
• 1 * P(1) = 1 * 0.2 = 0.2
• 2 * P(2) = 2 * 0.3 = 0.6
• 3 * P(3) = 3 * 0.25 = 0.75
• 4 * P(4) = 4 * 0.15 = 0.6

Step 5: Sum the Results from Step 4 (Σ [x * P(x)])

Finally, sum all the results obtained in the previous step. This sum is the mean (expected value) of the discrete probability distribution.

Example (Continuing from Step 4):

μ = 0 + 0.2 + 0.6 + 0.75 + 0.6 = 2.15

Therefore, the mean number of cars sold per day is 2.15.

Examples of Calculating the Mean

Let's walk through several examples to reinforce the concept and demonstrate its application in different scenarios.

Example 1: Coin Flips

Suppose you flip a fair coin three times. Let X be the number of heads. The probability distribution is as follows:

• X = 0 (TTT): P(0) = 1/8 = 0.125
• X = 1 (HTT, THT, TTH): P(1) = 3/8 = 0.375
• X = 2 (HHT, HTH, THH): P(2) = 3/8 = 0.375
• X = 3 (HHH): P(3) = 1/8 = 0.125

Calculate the mean number of heads.

1. Multiply each value by its probability:
   • 0 * P(0) = 0 * 0.125 = 0
   • 1 * P(1) = 1 * 0.375 = 0.375
   • 2 * P(2) = 2 * 0.375 = 0.75
   • 3 * P(3) = 3 * 0.125 = 0.375
2. Sum the results:

μ = 0 + 0.375 + 0.75 + 0.375 = 1.5

The mean number of heads is 1.5.

Example 2: Rolling a Fair Six-Sided Die

Consider rolling a fair six-sided die. The possible outcomes are 1, 2, 3, 4, 5, and 6, each with a probability of 1/6.

Calculate the mean value of the roll.

1. Multiply each value by its probability:
   • 1 * P(1) = 1 * (1/6) = 1/6
   • 2 * P(2) = 2 * (1/6) = 2/6
   • 3 * P(3) = 3 * (1/6) = 3/6
   • 4 * P(4) = 4 * (1/6) = 4/6
   • 5 * P(5) = 5 * (1/6) = 5/6
   • 6 * P(6) = 6 * (1/6) = 6/6
2. Sum the results:

μ = (1/6) + (2/6) + (3/6) + (4/6) + (5/6) + (6/6) = (1+2+3+4+5+6)/6 = 21/6 = 3.5

The mean value of the die roll is 3.5.

Example 3: Number of Defective Items

A manufacturing company produces items, and the number of defective items per batch follows a discrete probability distribution:

• X = 0 (0 defective items): P(0) = 0.6
• X = 1 (1 defective item): P(1) = 0.2
• X = 2 (2 defective items): P(2) = 0.1
• X = 3 (3 defective items): P(3) = 0.1

Calculate the mean number of defective items per batch.

1. Multiply each value by its probability:
   • 0 * P(0) = 0 * 0.6 = 0
   • 1 * P(1) = 1 * 0.2 = 0.2
   • 2 * P(2) = 2 * 0.1 = 0.2
   • 3 * P(3) = 3 * 0.1 = 0.3
2. Sum the results:

μ = 0 + 0.2 + 0.2 + 0.3 = 0.7

The mean number of defective items per batch is 0.7.

Example 4: Lottery Winnings

Consider a lottery where the possible winnings and their probabilities are as follows:

• X = $0: P(0) = 0.8
• X = $10: P(10) = 0.15
• X = $100: P(100) = 0.04
• X = $1000: P(1000) = 0.01

Calculate the mean winning amount.

1. Multiply each value by its probability:
   • 0 * P(0) = 0 * 0.8 = 0
   • 10 * P(10) = 10 * 0.15 = 1.5
   • 100 * P(100) = 100 * 0.04 = 4
   • 1000 * P(1000) = 1000 * 0.01 = 10
2. Sum the results:

μ = 0 + 1.5 + 4 + 10 = 15.5

The mean winning amount is $15.5.

Common Mistakes to Avoid

When calculating the mean of a discrete probability distribution, several common mistakes can lead to incorrect results. Being aware of these pitfalls can help ensure accuracy:

1. Forgetting to Multiply by Probability: A frequent error is simply summing the values of the random variable without multiplying each by its corresponding probability.
2. Incorrect Probabilities: Using incorrect probabilities or failing to verify that the sum of probabilities equals 1 can lead to a skewed mean.
3. Misidentifying Values of the Random Variable: Failing to correctly identify all possible values of the random variable or including impossible values can distort the result.
4. Arithmetic Errors: Simple arithmetic errors in multiplication or addition can lead to incorrect calculations. Double-check each step to avoid these mistakes.
5. Ignoring Zero Values: Not accounting for values of the random variable that have a value of zero, which can still affect the final mean if their probabilities are non-zero.

Practical Applications of the Mean

The mean of a discrete probability distribution has wide-ranging practical applications in various fields:

• Finance: In finance, the mean is used to calculate the expected return on an investment. For example, investors can use it to evaluate the potential profitability of stocks, bonds, or other assets.
• Insurance: Insurance companies use the mean to calculate expected payouts for policies. This helps them determine appropriate premium rates and manage risk.
• Quality Control: In manufacturing, the mean is used to monitor the number of defects in a production process. This helps identify when the process is deviating from acceptable standards.
• Marketing: Marketers use the mean to estimate the expected response rate to a marketing campaign. This helps them allocate resources effectively and measure the success of their efforts.
• Healthcare: In healthcare, the mean is used to analyze patient outcomes and evaluate the effectiveness of treatments. For example, it can be used to determine the average length of stay in a hospital or the average reduction in symptoms after a new medication.

Advanced Considerations

While the basic formula for calculating the mean is straightforward, there are some advanced considerations to keep in mind:

• Expected Value as a Long-Run Average: The mean of a discrete probability distribution represents the expected value in the long run. This means that if you repeat the experiment many times, the average of the outcomes will converge to the mean.
• Relationship to Variance and Standard Deviation: The mean is often used in conjunction with the variance and standard deviation to fully describe the distribution. The variance measures the spread of the data around the mean, while the standard deviation is the square root of the variance.
• Conditional Expectation: In some cases, you may need to calculate the conditional expectation, which is the expected value of a random variable given some condition.
• Continuous Probability Distributions: While this guide focuses on discrete probability distributions, the concept of the mean also applies to continuous probability distributions. However, the formula is slightly different, involving integration rather than summation.

The Role of Technology in Calculating the Mean

In modern data analysis, technology plays a crucial role in calculating the mean and other statistical measures. Spreadsheet software like Microsoft Excel and Google Sheets, as well as statistical software packages such as R, Python (with libraries like NumPy and SciPy), and SPSS, provide functions and tools to simplify these calculations. These tools not only automate the computation but also help in visualizing and analyzing the distribution.

For instance, in Excel, you can easily calculate the mean by entering the values of the random variable and their corresponding probabilities in separate columns and then using a formula that multiplies the values and probabilities and sums the results. Similarly, in Python, NumPy's array operations can efficiently perform these calculations.

Conclusion

Calculating the mean of a discrete probability distribution is a fundamental skill with broad applications across various fields. By following the step-by-step guide and avoiding common mistakes, you can accurately determine the expected value of a random variable. Whether you're analyzing financial investments, quality control processes, or marketing campaigns, understanding how to calculate the mean is essential for making informed decisions and drawing meaningful conclusions from data. The mean serves as a cornerstone in statistical analysis, providing a crucial measure of central tendency that helps to summarize and interpret complex datasets.