6.
When is median useful, as opposed to using the mean?
When the data is skewed irregularly, median is more useful.
For example, if we have a data set of {0,1,2,3,4,5,6,7,10000}, the median is 4, and the mean is 1114.2.
Since the data is most around 4, median is more useful in this situation.
7.
How do you calculate probability using the various rules?
For continuous random variable/probability, we can calculate probability of the standard form
P(X<x) by checking the cdf, since if F(x) is the cdf of a random variable X, the F(x)=P(X<x).
For P(X>x), we can transform it to 1-P(X<x), and then use cdf to find out P(X<x)
For P(x1<X<x2), we transform it to P(X<x2)-P(X<x1), and use cdf.
For discrete random variable/probability, we can calculate probability of the standard form
P(X<=x) by checking the cdf, since if F(x) is the cdf of a random variable X, the F(x)=P(X<=x).
For P(X>x), we can transform it to 1-P(X<=x)=1-F(x).