Question

1. What are PDFs and CDFs? What is the relationship between them? What is an inverse CDF? How would you use it to find the 95^th percentile of a distribution?

Answer 1

Answer:

PDF stands for probability density function. A function of a continuous random variable, whose integral across an interval gives the probability that the value of the variable lies within the same interval.

CDF stands for cumulative distribution function.  In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.

Difference between pdf and cdf:

Probability looks at probability at one point.

Cumulative is the total probability of anything below it.

As you can see in the diagram below, the cumulative is much greater than the just probability because it is the sum of many, and not just of one probabilities.As you can see in the diagram below, the cumulative is much greater than the just probability because it is the sum of many, and not just of one probabilities.

Inverse CDF:

The inverse distribution function for continuous variables F_x^-1(α) is the inverse of the cumulative distribution function (CDF). In other words, it's simply the distribution function F_x(x) inverted. The CDF shows the probability a random variable X is found at a value equal to or less than a certain x.

95th percentile:

By cumulative distribution function we denote the function that returns probabilities of X being smaller than or equal to some value x,

Pr(X≤x)=F(x).Pr(X≤x)=F(x).

This function takes as input xx and returns values from the [0,1][0,1] interval (probabilities)—let's denote them as pp. The inverse of the cumulative distribution function (or quantile function) tells you what x would make F(x) return some value pp,

F−1(0.95)=x.