Probability Distribution Calculator

How to Calculate Probability Distribution Statistics

Probability distributions are fundamental concepts in statistics that describe the likelihood of different outcomes in a random experiment. Key statistics such as the mean, variance, and standard deviation provide important insights into the characteristics of these distributions.

Formulas and Their Meanings

1. Mean (Expected Value): \(\mu = \sum_{i=1}^{n} x_i P(x_i)\)

• \(\mu\) is the mean or expected value
• \(x_i\) are the individual values
• \(P(x_i)\) are their respective probabilities

The mean represents the average or central tendency of the distribution.

2. Variance: \(\sigma^2 = \sum_{i=1}^{n} (x_i - \mu)^2 P(x_i)\)

• \(\sigma^2\) is the variance
• \(x_i\) are the individual values
• \(\mu\) is the mean
• \(P(x_i)\) are their respective probabilities

The variance measures the spread or dispersion of the distribution from its mean.

3. Standard Deviation: \(\sigma = \sqrt{\sigma^2}\)

• \(\sigma\) is the standard deviation
• \(\sigma^2\) is the variance

The standard deviation is the square root of the variance and provides a measure of dispersion in the same units as the original data.

Calculation Steps

1. Identify the values (\(x_i\)) and their corresponding probabilities (\(P(x_i)\)).
2. Calculate the mean by multiplying each value by its probability and summing the results.
3. Calculate the variance by subtracting the mean from each value, squaring the difference, multiplying by the probability, and summing the results.
4. Calculate the standard deviation by taking the square root of the variance.

Example

Let's consider a simple probability distribution:

x	P(x)
1	0.2
2	0.5
3	0.3

1. Calculate the mean:

\(\mu = (1 \times 0.2) + (2 \times 0.5) + (3 \times 0.3) = 0.2 + 1 + 0.9 = 2.1\)

2. Calculate the variance:

\(\sigma^2 = (1 - 2.1)^2 \times 0.2 + (2 - 2.1)^2 \times 0.5 + (3 - 2.1)^2 \times 0.3\)

\(\sigma^2 = 0.484 \times 0.2 + 0.01 \times 0.5 + 0.81 \times 0.3 = 0.0968 + 0.005 + 0.243 = 0.3448\)

3. Calculate the standard deviation:

\(\sigma = \sqrt{0.3448} \approx 0.5872\)

Visual Representation

This diagram illustrates the probability distribution from our example. The x-axis represents the values, and the y-axis represents their probabilities. The green line shows the trend of the distribution.