Introduction to Probability Distributions

Key Takeaways

this lecture is going to serve as an overview of what a probability distribution is and what some of its main characteristics are simply put a distribution shows the possible values a variable can take and how frequently they occur before we start let us introduce some important notation we use for the remainder of the course assume that upper case Y represents the actual outcome of an event and lower case Y represents one of the possible outcomes one way to denote the likelihood of reaching a particular outcome Y is P of y equals y we can also express it as P of Y for example upper case Y could represent the number of red marbles we draw out of a bag and lower case Y would be a specific number like 3 or 5 then we express the probability of getting exactly 5 red marbles as P of y equals 5 or P of 5 since P of Y expresses the probability for each distinct outcome we call this the probability function good job folks so probability distributions or simply probabilities measure the likelihood of an outcome depending on how often it is featured in the sample space recall that we constructed the probability frequency distribution of an event in the introductory section of the course we recorded the frequency for each unique value and divided it by the total number of elements in the sample space usually that is the way we construct these probabilities when we have a finite number of possible outcomes if we had an infinite number of possibilities then recording the frequency for each one becomes impossible because there are infinitely many of them for instance imagine you are a data scientist and want to analyze the time it takes for your code to run any single compilation could take anywhere from a few milliseconds to several days often the result will be between a few milliseconds and a few minutes if we record time in seconds we lose precision which is something to be avoided to do so we need to use the smallest possible measurement of time since every milli micro or even nanosecond could be split in half for greater accuracy no such thing exists in less than an hour from now we will talk in more detail about continuous distributions and how to deal with them now is the time to introduce some key definitions regardless of whether we have a finite or infinite number of possibilities we define distributions using only two characteristics mean and variance simply put the mean of the distribution is its average value variance on the other hand is essentially how spread out the data is we measure this spread by how far away from the mean all the values are the more dispersed the data is the higher its variance will be we denote the mean of a distribution with the Greek letter mu and it's variance with Sigma squared okay when analyzing distributions it is important to understand what kind of data we are dealing with population or sample data population data is the formal way of referring to all the data while sample data is just a part of it for example if an employer surveys an entire department about how they travel to work the data would represent the population of the department however this same data would also just be a sample of the employees in the whole company something to remember when using sample data is that we adopt different notations for the mean and variance we denote sample mean as x-bar and sample variance as s squared one flaw of variance is that it is measured in squared units for example if you are measuring time in seconds the variance would be measured in seconds squared usually there is no direct interpretation of that value to make further sense of variance we introduce a third characteristic of the distribution called standard deviation standard deviation is simply the positive square root of variance as you may suspect we denote it as Sigma when dealing with a population and as s when dealing with a sample unlike variance standard deviation is measured in the same units as the mean thus we can directly interpret it and is often preferable one idea which we will use a lot is that any value between mu minus Sigma and mu plus Sigma falls within one standard deviation away from the mean the more congested the middle of the distribution the more data falls within that interval similarly the less data that falls within the interval the more dispersed the data is fantastic it is important to know that a constant relationship exists between mean and variance for any distribution by definition the variance equals the expected value of the squared difference from the mean for any value we denote this as Sigma squared equals the expected value of y minus mu squared after some simplification this is equal to the expected value of Y squared minus mu squared as you will see in the coming lectures if we are dealing with a specific distribution we can find a much more precise formula if you found this video interesting and want to gain an edge in your career make sure to LIKE comment and 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