Probability Density Function With Example | Probability And Statistics Tutorial

2. What’s in it for you? What is Probability Density Function? Steps to find Probability Density Function Summarize density with histogram Parametric Density Estimation Non-Parametric Density Estimation • S

3. What is Probability Density Function?

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5. What is Probability Density Function? A function which defines the relationship between a random variable and its probability, such that you can find the probability of the variable using the function is called a Probability Density Function (PDF). Consider a variable x with a pdf of f(x) P(a) ≤ X ≤ P(b) f(x) P(b) b a x Probability Density Function

6. What is Probability Density Function? Continuous Random Variable Discrete Random Variable A variable which can only take in a value within a certain range. The value is usually within a certain distance of another finite value, eg : Sum of two dice A continuous random variable is one which can take on infinite different values within a range of values, eg : Height of a student

7. What is Probability Density Function? Consider the graph below which shows the rainfall distribution in a year in a city. The x-axis has the rainfall in inches and the y-axis has the probability density function. The probability of some amount of rainfall is obtained by finding area of curve on the left of it P(Y = 3) ≤ 0.5 f(x) 0.5 3 x Amount of rain

8. Steps to find Probability Density Function

9. 1. Summarize Density with Histogram We first convert the data into discrete form by plotting it as a histogram. A histogram is a graph which has categorical values on x-axis and bins of different heights which give us a count of the values in that category Histogram

10. 1. Summarize Density with Histogram The number of bins is important as It determines how may bars the histogram will have and their width. This will determine how our density will be plotted Histogram of previous data with bin size 4

11. 2. Parametric Density Estimation • To check if our histogram is a good fit to the function, we can: • Plot the density function and compare histogram shape • Compare samples of the function with real samples • Use a statistical test A probability density function is of many types. The shape of the histogram will help us determine which type of function it is. We can calculate the parameters associated with the function to get our density Probability density function is normal distribution

12. 3. Non-Parametric Density Estimation In cases where the shape of our histogram doesn’t match a common probability density function, or cannot be made to fit one, we calculate the density using all samples in our data using certain algorithms Random Distribution

13. 3. Non-Parametric Density Estimation • Commonly used terms for non-parametric estimation are : • Kernel Density Estimation : Uses a mathematical function to calculate and smooth probabilities so that their sum is always 1 • Smoothing Parameter (bandwidth) : Controls the number of sample which are used to estimate the probability for a new point • Basis Function: Used to control distribution of sample Probability density function for random distribution

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