Apply transformations

Given a random variable \(Y\) that is a function of a random variable \(X\), \(Y=u(X)\), we can use the distribution function technique to find the pdf of \(Y\). $$ \begin{align*} & G\left(y \right)=P \left(Y ≤ y \right)=P\left[u\left(X \right) ≤ y…

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Determine the sum of independent random variables (Poisson and normal)

Given \(X\) and \(Y\) are independent random variables, then the probability density function of \(X+Y\) can be shown by the equation below: $$ { f }_{ X+Y }\left( a \right) =\int _{ -\infty }^{ \infty }{ { f }_{ X…

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Define probability generating functions and moment generating functions

The probability generating function of a discrete random variable is a power series representation of the random variable’s probability density function as shown in the formula below: \(G\left( n \right) =P\left( X=0 \right) \ast n^{ 0 }+P\left( X=1 \right) \ast…

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Explain and calculate variance, standard deviation, and coefficient of variation

The variance of a discrete random variable is the sum of the square of all the values the variable can take times the probability of that value occurring minus the sum of all the values the variable can take times…

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Explain and calculate expected value, mode, median, percentile and higher moments

The expected value of a discrete random variable is the sum of all the values the variable can take times the probability of that value occurring as in the formula shown below: $$ E\left( X \right) =\sum { xp\left( x…

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Explain and apply the concepts of random variables

Given a random experiment, with sample space, \(S\), we can define the possible values of \(S\) as a random variable. Random variables can be discrete or continuous. A discrete random variable is a variable whose range of possible values is…

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