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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State Bayes Theorem and use it to calculate conditional probabilities

An extension of conditional probability is Bayes Theorem which can be written as: $$ P\left( A|B \right) =\frac { P\left( B|A \right) \ast P\left( A \right) }{ P\left( B \right) } $$ Example Given a population of students in which…

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Calculate probabilities using combinatorics, such as combinations and permutations

Permutations In probability, we may be interested in the possible arrangement of a set of objects. If we are interested in the order of the arrangement of the objects, we call this arrangement a permutation. Example: Consider flipping 2 coins….

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Define and calculate conditional probabilities

Given two events \(A\) and \(B\), the conditional probability of event \(A\) occurring, given that event \(B\) has occurred is the probability of event \(A\) and \(B\) occurring over the probability of event \(B\) occurring as shown in the formula…

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Define independence and calculate probability of independent events

Independent Events In probability, two events (\(A\) and \(B\)) are said to be independent if the fact that one event (\(A\)) occurred does not affect the probability that the other event (\(B\)) will occur Consider the experiment rolling a die…

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