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Introduction

The normal distribution, also known as the Gaussian distribution, is one of the most important probability distributions in statistics. It models many natural phenomena such as heights, weights, test scores, and errors in measurements. Its distinctive bell-shaped curve makes it easily recognizable.

Definition

A random variable X is normally distributed with mean μ and standard deviation σ if its probability density function (PDF) is:

f(x) = (1 / (σ√(2π))) * e-(x - μ)² / (2σ²)

This is denoted as:

X ~ N(μ, σ²)

Characteristics

• Symmetric about the mean μ
• Mean, median, and mode are all equal
• Bell-shaped curve
• Total area under the curve = 1
• Follows the empirical rule (68-95-99.7 rule)

Empirical Rule (68-95-99.7)

For a normal distribution:

• ~68% of values fall within 1 standard deviation of the mean
• ~95% fall within 2 standard deviations
• ~99.7% fall within 3 standard deviations

Standard Normal Distribution

When μ = 0 and σ = 1, the distribution is called the standard normal distribution and is denoted as:

Z ~ N(0, 1)

You can convert any normal variable to a standard normal using the Z-score:

Z = (X - μ) / σ

Applications

• Modeling measurement errors
• Statistical inference (e.g., confidence intervals, hypothesis testing)
• Standardized test scoring
• Quality control in manufacturing

Example

Suppose the heights of adult men are normally distributed with mean μ = 175 cm and standard deviation σ = 10 cm. What is the probability that a randomly chosen man is taller than 190 cm?

Z = (190 - 175) / 10 = 1.5

P(X > 190) = P(Z > 1.5) ≈ 0.0668

So there’s about a 6.68% chance a randomly selected man is taller than 190 cm.

Python Code Example

from scipy.stats import norm

Mean and standard deviation

mu = 175 sigma = 10

Probability of height greater than 190 cm

p = 1 - norm.cdf(190, loc=mu, scale=sigma) print("P(X > 190):", p)

Conclusion

The normal distribution is central to statistics due to the Central Limit Theorem, which states that the sum of many independent random variables tends to be normally distributed. Its properties make it a cornerstone in many statistical procedures, from hypothesis testing to regression analysis.

Introduction

The exponential random variable is a continuous probability distribution that describes the time between independent events occurring at a constant average rate. It is widely used in reliability engineering, queuing theory, survival analysis, and various stochastic processes.

Definition

If a random variable X follows an exponential distribution with parameter λ > 0, we write:

X ~ Exp(λ)

Here, λ is the rate parameter, which represents the average number of events per unit time. The mean time between events is 1/λ.

Probability Density Function (PDF)

f(x) = λ * e-λx   for x ≥ 0

f(x) = 0              for x < 0

The PDF shows that the probability decreases exponentially as x increases. The distribution is heavily right-skewed.

Cumulative Distribution Function (CDF)

F(x) = 1 - e-λx  for x ≥ 0

F(x) = 0            for x < 0

The CDF gives the probability that the time until the next event is less than or equal to x.

Mean and Variance

• Mean: E[X] = 1/λ
• Variance: Var(X) = 1/λ²

Key Property: Memorylessness

The exponential distribution is the only continuous distribution that is memoryless. That is:

P(X > s + t | X > s) = P(X > t)

This means the probability that the process lasts at least another t units of time does not depend on how much time has already passed.

Example

Suppose the average number of phone calls received by a call center is 4 per hour. Then the time between two consecutive calls is exponentially distributed with λ = 4.

- Probability that you wait less than 15 minutes (0.25 hours) for the next call:

P(X < 0.25) = 1 - e-λx = 1 - e-4 * 0.25 = 1 - e-1 ≈ 0.632

So there's about a 63.2% chance a call comes within the first 15 minutes.

Applications

• Modeling time between arrivals in a Poisson process
• Reliability of systems (e.g. lifespan of electronic components)
• Survival analysis in medical statistics
• Queuing systems (e.g., wait time until next customer)

Python Code Example

from scipy.stats import expon

Set lambda (rate) = 4 => scale = 1/lambda

scale = 1 / 4

Mean and variance

mean, var = expon.mean(scale=scale), expon.var(scale=scale) print("Mean:", mean) print("Variance:", var)

Probability of waiting less than 15 minutes (0.25 hours)

p = expon.cdf(0.25, scale=scale) print("P(X < 0.25):", p)

Conclusion

The exponential random variable is essential for modeling the timing of random events. Its simplicity and powerful properties—particularly memorylessness—make it foundational in probability theory, stochastic modeling, and real-world applications involving waiting times and failure rates.

Introduction

The uniform random variable is one of the simplest and most fundamental probability distributions. It models a situation in which all outcomes in a given interval are equally likely. It’s often used as a building block for other distributions and is essential in simulations and Monte Carlo methods.

Definition

A uniform random variable can be either discrete or continuous. The most common and widely used is the continuous uniform distribution.

A continuous uniform random variable X is said to be uniformly distributed over the interval [a, b], written as:

X ~ U(a, b)

This means that the probability of X falling anywhere in the interval [a, b] is equally likely.

Probability Density Function (PDF)

The PDF of a uniform distribution on [a, b] is defined as:

f(x) = 1 / (b - a),  for a ≤ x ≤ b

f(x) = 0,           otherwise

This flat or constant density function indicates that all values in the interval [a, b] are equally probable.

Cumulative Distribution Function (CDF)

The CDF, which gives the probability that X is less than or equal to a certain value x, is:

F(x) = 0               if x < a

F(x) = (x - a) / (b - a) if a ≤ x ≤ b F(x) = 1                  if x > b

Mean and Variance

For X ~ U(a, b):

• Mean: E[X] = (a + b) / 2
• Variance: Var(X) = (b - a)2 / 12

Example

Suppose a bus arrives at a stop every 20 minutes. If you arrive at a random time, your waiting time X is uniformly distributed between 0 and 20 minutes: X ~ U(0, 20).

– The probability that you wait less than 5 minutes:

P(X ≤ 5) = (5 - 0) / (20 - 0) = 5 / 20 = 0.25

– The average wait time:

E[X] = (0 + 20) / 2 = 10 minutes

Discrete Uniform Distribution

A discrete uniform distribution assigns equal probability to a finite set of n outcomes. For example, rolling a fair 6-sided die produces outcomes {1, 2, 3, 4, 5, 6}, each with probability 1/6.

If X is uniformly distributed over {x1, x2, ..., xn}, then:

P(X = xi) = 1 / n  for all i

Applications

• Random number generation
• Simulation and modeling in Monte Carlo methods
• Estimating probabilities in uniformly distributed data
• Game theory and lotteries

Python Code Example

from scipy.stats import uniform

Continuous uniform from a = 0 to b = 20

mean, var = uniform.mean(loc=0, scale=20), uniform.var(loc=0, scale=20) print("Mean:", mean) print("Variance:", var)

Probability of waiting less than 5 minutes

p = uniform.cdf(5, loc=0, scale=20) print("P(X ≤ 5):", p)

Conclusion

The uniform random variable is a foundational concept in probability and statistics. It models situations with equal likelihood and serves as a simple but powerful tool in data analysis and simulation. Whether dealing with continuous ranges or finite discrete sets, understanding uniform distributions helps establish deeper intuition for randomness and fair systems.

Introduction

The negative binomial distribution is a fundamental probability distribution used in statistics to model the number of independent Bernoulli trials needed to achieve a fixed number of successes. It generalizes the geometric distribution, which models the number of trials until the first success.

This distribution is especially useful in situations where you are interested in counting the number of attempts needed to observe a specific number of successful outcomes, with each trial having the same probability of success.

Definition

Let X be a random variable representing the number of trials needed to get r successes in a sequence of independent Bernoulli trials, each with probability of success p. Then X follows a Negative Binomial Distribution.

The number of failures before the r-th success is what is often modeled. The probability mass function (PMF) is:

P(X = x) = C(x + r - 1, r - 1) * p^r * (1 - p)^x

for x = 0, 1, 2, ...

Here, X counts the number of failures before the r-th success, and C(n, k) denotes the binomial coefficient:

C(n, k) = n! / (k! * (n - k)!)

Parameters

• r: Number of desired successes (a positive integer)
• p: Probability of success on each trial (0 < p < 1)
• X: Number of failures before achieving r successes

Mean and Variance

If X follows a Negative Binomial Distribution with parameters r and p, then:

• Mean (Expected value): E[X] = r * (1 - p) / p
• Variance: Var(X) = r * (1 - p) / p²

Special Case: Geometric Distribution

The geometric distribution is a special case of the negative binomial distribution when r = 1. In that case, the negative binomial distribution simplifies to counting the number of failures before the first success.

Example

Suppose you are rolling a die, and you define success as rolling a 6 (p = 1/6). What is the probability that you roll the die 10 times and get the 3rd success on the 10th roll?

First, you must have had x = 7 failures before the 3rd success (since 10 – 3 = 7), and r = 3. Plug into the formula:

P(X = 7) = C(7 + 3 - 1, 3 - 1) * (1/6)^3 * (5/6)^7
     = C(9, 2) * (1/216) * (78125 / 279936)

Calculate and simplify for the numerical result.

Applications

• Modeling the number of accidents before a fixed number of safe days
• Predicting the number of failed transactions before reaching a success quota
• Call center analytics (e.g., number of calls before getting 5 successful sales)
• Quality assurance and manufacturing defects tracking

Python Code Example

from scipy.stats import nbinom

r = 3        # number of successes p = 1/6      # probability of success x = 7        # number of failures

Probability of exactly 7 failures before 3rd success

prob = nbinom.pmf(x, r, p) print(f"P(X = {x}) = {prob:.6f}")

Mean and variance

mean = nbinom.mean(r, p) var = nbinom.var(r, p) print(f"Mean: {mean}, Variance: {var}")

Conclusion

The negative binomial distribution is a versatile and powerful tool in probability, especially useful for modeling events where multiple successes are required over a sequence of trials. It generalizes the geometric distribution and has wide applications in quality control, economics, public health, and more.

Understanding its structure, formula, and behavior allows analysts and statisticians to model uncertainty in a wide range of real-world processes where success isn’t guaranteed on the first few tries.

Introduction

In probability theory, the geometric random variable is one of the most fundamental discrete random variables. It models the number of Bernoulli trials needed to achieve the first success. This makes it particularly useful in scenarios involving repeated, independent attempts at something with constant success probability—such as flipping a coin until heads appears, or testing a machine until it works.

Definition

A geometric random variable is a type of discrete random variable that represents the number of independent Bernoulli trials required to get the first success. Each trial results in either a success (with probability p) or a failure (with probability 1 - p).

There are two common definitions of the geometric random variable, depending on how you count the trials:

1. Definition 1: Number of trials until the first success (includes the success itself): X = 1, 2, 3, ...
2. Definition 2: Number of failures before the first success: Y = 0, 1, 2, ...

For this article, we will focus on Definition 1, which is more widely used.

Probability Mass Function (PMF)

If X is a geometric random variable with success probability p, then:

P(X = x) = (1 - p)x - 1 * p, for x = 1, 2, 3, ...

Explanation:

• (1 - p)x - 1: The probability of failing the first x - 1 times
• p: The probability of succeeding on the x-th trial

Cumulative Distribution Function (CDF)

The cumulative probability that the first success occurs on or before trial x is:

P(X ≤ x) = 1 - (1 - p)x

This tells us the likelihood of seeing a success within the first x trials.

Mean and Variance

Let X ~ Geometric(p). Then:

• Expected value (Mean): E[X] = 1 / p
• Variance: Var(X) = (1 - p) / p²

This implies that the rarer the success (smaller p), the longer (on average) you’ll wait to see the first success.

Memoryless Property

The geometric distribution is unique among discrete distributions for having the memoryless property:

P(X > m + n | X > m) = P(X > n)

This means the probability of the process lasting more than m + n trials, given that it has already lasted m trials without success, is independent of m. Past failures do not affect future probabilities.

Examples

Example 1: Coin Toss

Suppose you flip a fair coin (p = 0.5) and want to find the probability that the first head appears on the 3rd flip:

P(X = 3) = (1 - 0.5)^2 * 0.5 = 0.125

There is a 12.5% chance that you will get the first head on the third toss.

Example 2: Defective Machine

Imagine a machine produces items, and each item has a 10% chance of being defective (p = 0.1). Let X be the number of items tested until the first defective one is found.

Expected number of items to test: E[X] = 1 / 0.1 = 10
So, on average, you expect to find a defective item after testing 10 items.

Relation to Bernoulli and Binomial Distributions

• Bernoulli trial: a single trial with success/failure outcome
• Binomial distribution: counts number of successes in n trials
• Geometric distribution: counts number of trials until first success

The geometric distribution can be seen as a “waiting time” model for the first success.

Applications

• Reliability Engineering: Time until first failure in a system
• Quality Control: Number of items tested before finding a defect
• Computer Science: Iterations until a loop condition is met
• Finance: Modeling rare events like defaults or crashes

Python Code Example

import numpy as np

from scipy.stats import geom

Parameters

p = 0.3  # probability of success

PMF: probability first success on 4th trial

x = 4 prob = geom.pmf(x, p) print(f"P(X = {x}) = {prob:.4f}")

Expected value

mean = geom.mean(p) print(f"Expected value: {mean}")

Variance

var = geom.var(p) print(f"Variance: {var:.2f}")

Conclusion

The geometric random variable is an essential concept in probability, modeling the number of attempts before success in repeated independent trials. It’s particularly useful in reliability studies, simulation modeling, and stochastic processes. Its simplicity, memoryless property, and real-world relevance make it one of the foundational tools in both theoretical and applied statistics.