Problems for Cumulative Distribution Function

Earlier we discussed the introduction of a continuous random variable and cumulative distribution function. Please see this article if you haven’t read it about already. Now we will give some problems related to Cumulative distribution function so you can have a deeper understanding of this topic. These problems are adapted from the following textbook: “Probability and Random Processes by Scott Miller 2nd Edition.” Problem-1: Which of the following mathematical functions could be the CDF of some random variable? Remember: To be a CDF of some random variable, the function \(F_{X}(x)\) must start at zero when \(x = -\infty\), end at one...

Random Variables

A random variable is a real valued function of the elements of sample space S which maps each possible outcome  to a real number specified by some rule. Random variables are denoted by capital Alphabet like X. If the random variable takes on finite or countably an infinite number of values then the random variable is a discrete random variable and if it takes uncountably infinite values then it is a continuous random variable. As we said earlier that the mapping is done by some rule; Probability mass function \(P_X(x)\) defines that rule. \(P_X(x)\) of a random variable is a function that...

Baye’s Rule

In Probability theory, we came across some problems in which it is difficult to compute conditional probability directly. To explain this let us consider two events A and B, then their joint probability is given as: \(\begin{eqnarray}P(A,B) = P(A|B)P(B) \label{eqA}\end{eqnarray}\) and we know that: \(\begin{eqnarray}P(B|A) = \frac{P(A,B)}{P(A)} \label{eqB}\end{eqnarray}\) then using \eqref{eqA} and \eqref{eqB} we get: \(\begin{eqnarray}P(B|A) = \frac{P(A|B)P(B)}{P(A)} \label{bayes}\end{eqnarray}\) Equation \eqref{bayes} is called Baye’s Rule and is useful for calculating certain conditional probabilities as in many problems it is difficult to compute P(B|A) directly, hence we use Baye’s Rule to solve such problems. A few examples are given below:...