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Variance and standard deviation of a discrete random variable

We learn how to calculate the mean and standard deviation of a discrete random variable. The concept of a random variable is explained, along with methods to calculate its expected value (mean) and measure its spread (variance and standard deviation). A practical example makes the concept easier to understand.

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• why we didn't divide by n or (n - 1)?
• i thought that the mean was the average of the sum. how come the mean of the discrete random variable is only the sum of the x(p(x)) instead of sum:x(p(x)) / number of values of x? I hope that makes sense.
• The expected value is sigma xp(x) by definition. What this implies if there are three numbers let say 1, 5, 10, and three number have equally likely chance of occurring:

then the expected value is (1+5+10)/3 = 16/3 = 5.33...

If the probabilty the values occurring are different then you would have to use xp(x). Let now say 1 occurs with 0.5 chance, 10 with chance of 0.2 and 5 with chance of 0.3 . Then the expected value is 0.5(1)+0.3(5)+0.2(10)= 3.4.

Note that mean and expected value are the same thing. It is just we extending concept of mean or in other words expected value for various probability mass function where each event does not have the same chance of occurring.
• What is the difference between variance and standard deviation? Why is standard deviation the square root of the variance? Thanks!
• The variance is an indicator of the dispersion but doesn't carry any immediate information about it (for instance, how could you interpret a variance of 1.19 from a random variable in comparison with a variance of 2.34 from another r.v.?).

Standard deviation allows you to "standardize" the dispersion for large number of samples (or initially based on normal distribution): if your std is 1.09 and your mean is 2.1, you can say that 68% of your values are expected to be between 2.1-1.09 and 2.1+1.09 (mean + 1 std) for instance.

Basically (and quite naively), std is a way to standardize the value given by the variance.
• At , how is for standard deviation to be intuitive? Given the way it is calculated, I think standard deviation is similar to the mean of deviation. But how can it be reasonable intuitively?
• Just a quick question, why do we have to time P(X) again when calculating Standard Deviation? This was done already when calculating E(X) so the mean 2.1 should be weighted already no?
• I think a good way to understand 'weight' is with the concept of frequency. With the random variable X, 0 occurs 10% of the time. So, there were 100 data points/ experiments, we would estimate 10 of them to be zero. Of course, we do not have the number of data points, but we do have the frequency that 0 will probably occur.

Long story short, we cannot ignore how often a data point shows up, otherwise we are ignoring a big portion of the data.

Forgive me if that was confusing. If you need clarification, let me know.
• So basically, the formula of finding variance of a discrete random variable is

X= random variable
P(X)= probability of random variable
Σ=sum
σ^2= variance
µ=mean

``σ^2 =Σ[X-µ ]^2 ⋅ P(X)  variance is equal to the sum of squared difference between X(respectively) and the mean(µ), then we multiply it with the P(X) the X's probability_.``

To find the standard deviation(σ), we simply just have to take square root of both side,( usually do it after found your variance):

√（ σ^2） =√（Σ[X-µ ]^2 ⋅ P(X)）

Feel free to correct me if I have made any mistake here, since im just another learner as well, cant be 100% right.
• What I love about Sal is that he explains the concept behind every equation or method so we wouldn't have to just memorize it.
But he didn't do so this time with the VAR equation. Can anyone explain it?