Probability Distribution – Essays UK

Key Probability Distribution Formulas

To understand how probability distributions work mathematically, it is essential to know the core functions and formulas used to describe them. 

Probability Mass Function (PMF)

The Probability Mass Function (PMF) is used for discrete probability distributions. It provides the probability that a discrete random variable takes on a specific value.

Formula: P(X = x) = f(x)

Where:

• X = discrete random variable
• x = specific value of X
• f(x) = probability of X taking the value x
  

The PMF satisfies two important conditions:

1. f(x) ≥ 0 for all x
2. Σ f(x) = 1
   

Example: In a binomial distribution with n = 3 and p = 0.5, the PMF gives the probability of getting 0, 1, 2, or 3 successes.

Probability Density Function (PDF)

The Probability Density Function (PDF) applies to continuous probability distributions. Instead of assigning a probability to individual values, it defines a curve where the area under the curve within an interval represents the probability.

Formula: P(a ≤ X ≤ b) = ∫ from a to b f(x) dx

Where:

• f(x) = PDF of the continuous random variable X
• The total area under f(x) from −∞ to +∞ equals 1
  

Example: For a normal distribution, the PDF produces the well-known bell-shaped curve, showing how data cluster around the mean.

Cumulative Distribution Function (CDF)

The Cumulative Distribution Function (CDF) gives the probability that a random variable takes a value less than or equal to a particular number. It applies to both discrete and continuous distributions.

• Formula: F(x) = P(X ≤ x)
• For discrete distributions: F(x) = Σ f(t) for all t ≤ x
• For continuous distributions: F(x) = ∫ from −∞ to x f(t) dt

The CDF increases monotonically from 0 to 1 as x moves from the smallest to the largest possible value.

Example: In a uniform distribution between 0 and 1, F(0.4) = 0.4, meaning there is a 40% probability that X ≤ 0.4.

Mean & Variance Of Distributions

The mean and variance summarise a probability distribution’s central tendency and spread.

• E(X) = Σ x·P(x) (for discrete)
• E(X) = ∫ x·f(x) dx (for continuous)

The mean shows the long-run average outcome of a random variable.

• • Var(X) = Σ (x − μ)²·P(x) (for discrete)
  • Var(X) = ∫ (x − μ)²·f(x) dx (for continuous)

Variance measures how much the outcomes deviate from the mean.

Probability Distributions In Excel & SPSS

Modern statistical tools like Microsoft Excel and IBM SPSS make it easy to calculate, visualise, and interpret probability distributions without complex manual formulas. 

How To Use Excel Functions For Probability Distributions

Excel provides built-in functions for different types of probability distributions. Here are some important functions. 

NORM.DIST(x, mean, standard_dev, cumulative)

Used to calculate probabilities in the normal distribution. Setting cumulative = TRUE gives the cumulative probability, while setting it to FALSE returns the probability density.

BINOM.DIST(number_s, trials, probability_s, cumulative)

Calculates probabilities for the binomial distribution, such as the likelihood of a certain number of successes in fixed trials.

POISSON.DIST(x, mean, cumulative)

Computes probabilities for the Poisson distribution, useful for modelling rare events within a fixed time or space.

How To Generate Probability Plots In SPSS

SPSS provides a user-friendly interface for analysing probability distributions through its Descriptive Statistics and Graphs tools. Researchers can compute important statistics and visualise how data align with theoretical distributions.

1. Open your dataset in SPSS.
2. Go to Analyse > Descriptive Statistics > Explore.
3. Move the target variable into the Dependent List box.
4. Click on Plots and choose Normality plots with tests.
5. Run the analysis to view histograms, Q-Q plots, and Kolmogorov–Smirnov/Shapiro–Wilk tests.

Example Output Interpretation

• A bell-shaped histogram suggests data follow a normal distribution.
• In a Q-Q plot, points that closely align with the diagonal line indicate normality.
• Significance values (p > 0.05) in normality tests imply the data do not significantly deviate from a normal distribution.

Frequently Asked Questions