Describing probability distributions and probability distributions with multiple variables
This week you will learn about different measures to describe probability distributions and any dataset. These include measures of central tendency (mean, median, and mode), variance, skewness, and kurtosis. The concept of the expected value of a random variable is introduced to help you understand each of these measures. You will also learn about some visual tools to describe data and distributions. In lesson 2, you will learn about the probability distribution of two or more random variables using concepts like joint, marginal, and conditional distributions. You will end the week by learning about covariance: a generalization of variance to two or more random variables.
Describing Distributions
Expected Value
Mean is the sum of all elements divided by the total number of elements.
We can think of the expected value as the same as the mean.
You play a game with a friend. The game is as follows: You throw a fair coin. If it lands heads you win 10 dollars, otherwise you win nothing. What's the maximum amount you should be willing to pay to play this game?
$5
This is the average amount you'd win if you played the game many times.
What's the maximum amount you should be willing to play this new game, where you flip 3 coins and win a dollar for each heads-up get?
$1.50
This is the average or expected amount of money you'd win if you played the game many times.
Depending on the weights given (a probability of each case), the mean changes.
We use integrals to calculate and get the mean of the continuous random variables.
With the uniform distribution, the average is the middle of points a and b (two endpoints).
When the distribution is not uniform or symmetrical, the mean would not be in the middle.
Rather, the median will be the middle point.
Other measures of central tendency: median and mode
Mean doesn’t always work as the central point of the distribution due to outliers.
Outliers are the points that are way off from all the other points.
To overcome the issue with outliers, we use the median, which is the middle point of all the given data spread out in one line.
Mode is the number that shows up the most in the data or the maximum point in the distribution.
If we have multiple modes in the distribution, we call it multimodal distribution.
Expected value of a Function
To get the expected value of a function, we multiply the probabilities of variables with functions of variables and sum them up.
So here with the function of squaring variables (on the last slide), we square the outputs and divide by 6 to get the expected value (mean).
On the formula, we can group the constant and take out the constant from the expected value like on the last slide.
This becomes like the linear function (ax + b).
Sum of expectations
Game: You flip a coin. If heads, you win $1, otherwise you win nothing. Then you roll a dice. You win the amount you roll. What are your expected winnings for this game?
$4
What is the expected number of correct assignments when we have 8 billion people and 8 billion unique names in the bag and hand out the names to everyone?
1
The sum of expectations is simply adding the average of each expectation and if everything is added, they add up to 1.
All the information provided is based on the Probability & Statistics for Machine Learning & Data Science | Coursera from DeepLearning.AI
