Stats 1 week 12

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Probability Distributions Summary

Continuous Random Variable

A continuous random variable is a variable that can take an infinite number of values within a given range. Unlike discrete random variables, which have specific values, continuous random variables are described by a probability density function (PDF).

Example: The height of students in a class is a continuous random variable. It can take any value within a certain range.

Probability Distribution Curve

The probability distribution curve represents the probability density function (PDF) of a continuous random variable. The area under the curve between two points represents the probability that the variable falls within that range.

Example: If you want to find the probability that a student's height is between 150 cm and 160 cm, you would use the PDF to calculate the area under the curve between these two points.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a continuous random variable is a function that shows the probability that the variable will take a value less than or equal to a given point. It is the integral of the PDF from negative infinity to that point.

Example: To find the probability that a student's height is less than 160 cm, you would use the CDF.

Uniform Distribution

The uniform distribution is a type of continuous probability distribution where all outcomes are equally likely within a certain range. The PDF of a uniform distribution is constant between the minimum and maximum values.

Example: If you roll a fair die, the probability of each outcome (1 through 6) is the same. This can be modeled using a uniform distribution.

Standard Uniform Distribution

The standard uniform distribution is a special case of the uniform distribution where the minimum value is 0 and the maximum value is 1. The PDF is 1 within this range and 0 outside of it.

Example: Randomly selecting a number between 0 and 1 can be modeled using a standard uniform distribution.

Cumulative Distribution of Uniform Distribution

The CDF of a uniform distribution increases linearly from 0 to 1 as the variable moves from the minimum to the maximum value. For a standard uniform distribution, the CDF is equal to the value of the variable itself.

Example: To find the probability that a randomly selected number between 0 and 1 is less than 0.5, you would use the CDF of the standard uniform distribution.

Non-Uniform and Triangular Distribution

Non-uniform distributions have varying probabilities for different outcomes. The triangular distribution is a type of non-uniform distribution with a triangular-shaped PDF. It is defined by a minimum value, a maximum value, and a mode (the peak of the triangle).

Example: The time it takes to complete a task might be modeled using a triangular distribution if you know the minimum, maximum, and most likely times.

Exponential Function

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. Its PDF decreases exponentially and is defined by a rate parameter λ.

Example: The time between arrivals of buses at a bus stop can be modeled using an exponential distribution.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution characterized by its bell-shaped curve. It is defined by its mean (μ) and standard deviation (σ). The PDF of a normal distribution is given by the formula:

        f(x) = (1 / (σ√(2π))) * exp(-0.5 * ((x - μ) / σ)^2)
    

Example: The distribution of heights of adult men in a population can be modeled using a normal distribution.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1. It is often used to standardize other normal distributions.

Example: Converting a normal distribution of test scores to a standard normal distribution allows for easier comparison of scores.

Standardizing a Normal Random Variable

To standardize a normal random variable, you subtract the mean and divide by the standard deviation. This process converts the variable to a standard normal distribution. The formula is:

        Z = (X - μ) / σ
    

Example: If a test score is normally distributed with a mean of 70 and a standard deviation of 10, a score of 85 can be standardized using the formula to find its corresponding Z-score.

This summary provides an overview of key concepts related to continuous random variables and various probability distributions. Understanding these concepts is crucial for statistical analysis and probability theory.

In the context of normal distribution, only basic questions will be asked, such as standardizing it or finding probabilities using the 68-95-99.7 rule. If you are familiar with these concepts, you should be able to solve them easily. In the stats revision PDF, I have explained these two concepts, which should be more than enough.

Some impotant question for week 11 and 12