Random Variable Collection

• A random variable is a numerical outcome of a random phenomenon, which can be classified into two types: discrete and continuous.

Discrete random variable

• Bernoulli RV - eg. x=0, x=1 P(x=0)=q, P(x=1)=p p+q=1

• Binomial RV (Bin(m,p)) - Represents the number of successes in a fixed number of independent Bernoulli trials, with parameters m (number of trials) and p (probability of success on each trial). For example, if we conduct 10 trials, the random variable X could take values from 0 to 10, where P(X=k) is calculated using the binomial formula.

• Negative Binomial Variables (NB) - NB1 (m,q) number of trials until the m-th success occurs, where q is the probability of failure on each trial. The probability mass function is given by P(X=k) = \binom{k+m-1}{m-1} p^m q^k, where k is the number of failures before the m-th success. k= m, m+1, m+2,…..NB2- the number of fails before the m-th success k=0,1,2,3,…..[ P(k) = \binom{m+k-1}{m-1} = \frac{(m+k-1)!}{(m-1)!k!} ].This represents the number of ways to choose (m-1) successes from (m+k-1) total trials, where k represents the number of additional trials before achieving the m-th success.

• Poisson distribution: A probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. k=0,1,2,3,…..P(X = k) = \frac{e^{-\lambda} \lambda^{k}}{k!}

  Where:

  • P(X = k) = the probability of k events in the interval

  • e = Euler's number (approximately equal to 2.71828)

  • \lambda = the average number of events in the interval

  • k = the actual number of events that occur

  • k! = k factorial (the product of all positive integers up to k)