Continuous or Discrete ?

Continuous and Discrete Variable 

We will start quantitative analysis with simple description of continuous and discrete random variables, and their difference.

A continuous random variable (X) has an infinite number of values within an interval:






Whereas a discrete random variable (X) assumes a value among a finite set including X1, X2, X3 and so on. The probability function is expressed by:

In financial market risk analysis, we will be working with both discrete and continuous random variables. Discrete in the sense that sometime we will be discussion about bonds which will have one of the several letter ratings (AAA,AA,A, BBB,BB etc) and continuous random variables like return of the stock index. However when we are talking about probability in continuous variable we should keep in mind that probability of any specific value occurring is zero.  So we should give the probability 'in range' for a continuous variable: For example: In the return on a stock market index over the next year we can talk about the probability of the index return being between 6% and 7%, but talking about the probability of the return being exactly 6.001%  or exactly 6.002% is meaningless.

Probability density function, The cumulative distribution function (CDF) and the inverse CDF 

The probability density function answers a 'local' question: If the random variable is discrete, the PDF is the probability the variable will assume an exact value of x, i.e., PMF: P(X=x). If the variable is continuous, the pdf tell us the likelihood of outcomes occurring on an interval between any two points.

The cumulative density function (CDF) associates with a PMF or PDF (i.e, the CDF can apply to either a continuous or  random variables). The CDF gives the probability the random variable will be less than, or equal to, some value.

If you want to know more about CDF please refer this video

Inverse cumulative distribution function
If F(x) is a cumulative distribution function, then we define F^-1(p), the inverse cumulative distribution, as follows:

The inverse cumulative distribution function is also called the quantile function. So this might be one of the reason Value at Risk (VaR) is also defined as quantile function;. For example: 95% VaR is the inverse CDF for p=5% or p95%