Computing Probability Using Binomial Distribution Formula
Experiments Modeled by Binomial Distribution
Applications of Binomial Distribution
Examples of Binomial Distribution
Probability - Poisson Distribution
Introduction to Poisson Distribution
Characteristics of Poisson Distribution
Formula for Poisson Distribution
Mean and Variance of Poisson Distribution
Applications of Poisson Distribution
Probability - Poisson Distribution (contd.)
Computing Probability Using Poisson Distribution Formula
Relation Between Poisson and Binomial Distribution
Examples of Poisson Distribution
Exponential Distribution
Relationship Between Poisson and Exponential Distribution
Probability - Uniform Distribution
Introduction to Uniform Distribution
Characteristics of Uniform Distribution
Probability Density Function for Uniform Distribution
Mean and Variance of Uniform Distribution
Applications of Uniform Distribution
Probability - Uniform Distribution (contd.)
Computing Probability Using Uniform Distribution Formula
Examples of Uniform Distribution
Triangular Distribution
Applications of Triangular Distribution
Examples of Triangular Distribution
Probability - Standard Deviation
Interpreting Standard Deviation
Measures the dispersion of data points around the mean
How spread out the data is from the average
Larger standard deviation indicates greater variability
Can be thought of as the average distance between each data point and the mean
Probability - Standard Deviation (contd.)
How to compute standard deviation manually
Steps:
Compute the mean of the data set
Calculate the difference between each data point and the mean
Square each difference
Sum up the squared differences
Divide by the number of data points
Take the square root of the result to get the standard deviation
Example:
Data set: 2, 4, 6, 8, 10
Mean = (2 + 4 + 6 + 8 + 10) / 5 = 6
Difference from mean: -4, -2, 0, 2, 4
Squared differences: 16, 4, 0, 4, 16
Sum of squared differences: 40
Standard deviation = √(40/5) = 2
Probability - Standard Deviation (contd.)
Standard deviation for a sample versus population
Population standard deviation (σ) is used when the entire population is considered
Sample standard deviation (s) is used when only a portion of the population is considered
The formulas for population and sample standard deviation are slightly different
The sample standard deviation formula uses n-1 instead of n in the denominator to account for the smaller sample size
Formula:
σ = √(Σ(xᵢ - μ)² / N) for population
s = √(Σ(xᵢ - x̄)² / (n-1)) for sample
Example:
Data set: 2, 4, 6, 8, 10
Sample mean = (2 + 4 + 6 + 8 + 10) / 5 = 6
Sample standard deviation = √((2-6)² + (4-6)² + (6-6)² + (8-6)² + (10-6)²) / (5-1) = √(8/4) = √2 = 1.41
Probability - Standard Deviation (contd.)
Interpretation of standard deviation
Approximately 68% of data falls within one standard deviation of the mean
Approximately 95% falls within two standard deviations
Approximately 99.7% falls within three standard deviations
Used to compare data sets and determine their spread and variability
Can help identify outliers or unusual data points
Example:
Data set 1: 2, 4, 6, 8, 10
Standard deviation = 2
Data set 2: 1, 3, 5, 7, 9
Standard deviation = 2
Both data sets have the same standard deviation, but the actual values are different. This shows that the variability within each data set is the same.
Probability - Normal Distribution
Introduction to normal distribution
Bell-shaped curve that represents a continuous probability distribution
It is symmetrical around the mean
Mean and median of a normal distribution are equal
Many real-world phenomena can be modeled using the normal distribution
Example:
Height of people in a population can be modeled using a normal distribution. The mean height could be the average height of the population, and the standard deviation could represent how spread out the heights are.
Probability - Normal Distribution (contd.)
Characteristics of normal distribution
Follows a specific mathematical formula known as the probability density function
The total area under the curve is equal to 1
The curve is symmetric and centered around the mean
Bell-shaped with thin tails
Mean and standard deviation completely determine the shape of the distribution
Equation for the standard normal distribution:
f(z) = 1 / √(2π) * e^(-z²/2) where z is the z-score
Probability - Normal Distribution (contd.)
Standard normal distribution
A normal distribution with mean 0 and standard deviation 1
Useful for calculating probabilities using z-scores
Z-score is calculated by subtracting the mean from a data point and then dividing by the standard deviation
Formula for z-score:
z = (x - μ) / σ
Example:
If the mean height of a population is 160 cm and the standard deviation is 5 cm, a height of 170 cm would have a z-score of (170 - 160) / 5 = 2. A z-score of 2 means that the height is 2 standard deviations above the mean.
Probability - Normal Distribution (contd.)
Computing area under the normal curve
Area under the curve represents the probability of an event occurring
Calculating the area under the curve involves finding the cumulative probability up to a certain z-score or between two z-scores
Can be done using tables, software, or calculators
Example:
Find the probability of getting a z-score less than -1.5 using a standard normal distribution table. Look up the value for -1.5 in the table, which is 0.0668. This means that there is a 6.68% chance of getting a z-score less than -1.5.
Probability - Normal Distribution (contd.)
Finding probability using z-score
Finding the probability of an event occurring within a certain range
Can be done by finding the cumulative probability from one z-score to another
Can also be done by finding the difference between two cumulative probabilities
Example:
Find the probability of getting a z-score between -1.5 and 1.5. Start by finding the individual probabilities for -1.5 and 1.5 using the normal distribution table. The probability for -1.5 is 0.0668 and for 1.5 is 0.9332. The probability of the event occurring between these two z-scores is the difference between the two probabilities: 0.9332 - 0.0668 = 0.8664.
Probability - Normal Distribution (contd.)
Normal distribution applications
Used in hypothesis testing
Used in quality control processes
Used in forecasting and prediction models
Widely applicable in fields such as finance, economics, biology, and psychology
Provides a useful tool for analyzing and understanding data
Examples:
Predicting stock prices based on historical trends
Analyzing test scores of students to determine if there is a significant difference between two groups
Determining the probability of a disease occurring in a certain population based on risk factors
Probability - Standard Deviation
Standard deviation and the normal distribution
The 68-95-99.7 rule and standard deviations
Calculating the standard deviation given a set of data
Interpreting the standard deviation
Standard deviation in real-life applications
Probability - Standard Deviation (contd.)
Comparing standard deviations of data sets
Standard deviation and variability in data
Standard deviation and hypothesis testing
Standard deviation and risk assessment
Standard deviation in financial analysis
Probability - Standard Deviation (contd.)
Calculating the standard deviation for grouped data
The grouped frequency distribution formula for standard deviation
Weighted standard deviation for weighted data
Application of standard deviation in quality control
Examples of calculating standard deviation for different data sets
Probability - Normal Distribution
Introduction to the concept of normal distribution
The bell curve and normal distribution curve
Characteristics of a normal distribution
Probability density function for normal distribution
Central Limit Theorem and normal distribution
Probability - Normal Distribution (contd.)
Standardizing normal distribution using z-scores
Calculating cumulative probabilities for normal distribution
Using the empirical rule to estimate probabilities
Calculating probabilities using standard normal distribution table
Examples of normal distribution in real-life situations
Probability - Normal Distribution (contd.)
Standardizing and finding probabilities for different data sets
Calculating the area under the normal curve
Using z-scores to calculate percentiles
Calculating probabilities for specific z-scores
Applications of normal distribution in statistics and data analysis
Probability - Binomial Distribution
Introduction to binomial distribution
Characteristics of the binomial distribution
The binomial random variable and binomial experiments
Formula for calculating probabilities in binomial distribution
Example of a binomial distribution problem
Probability - Binomial Distribution (contd.)
Mean and variance of the binomial distribution
Finding the standard deviation for binomial distribution
Calculating probability using the binomial distribution formula
Applications of binomial distribution in real-life scenarios
Examples of situations that can be modeled using binomial distribution
Probability - Poisson Distribution
Introduction to Poisson distribution
Characteristics of the Poisson distribution
Formula for calculating probabilities in Poisson distribution
Mean and variance of the Poisson distribution
Examples of situations modeled by the Poisson distribution
Probability - Poisson Distribution (contd.)
Computing probabilities using the Poisson distribution formula
Relationship between the Poisson and binomial distribution
Applications of the Poisson distribution in various fields
The Poisson distribution in queuing theory
Examples of problems solved using the Poisson distribution