In this section we learn how to find the , mean, median, mode, variance and standard deviation of a discrete random variable.
We define each of these parameters:
It is worth spending a bit of time on this section as all that is taught here applies to all discrete random variable probability distributions, such as the Binomial Distribution as well as the Poisson Distribution.
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Given a discrete random variable \(X\), its mode is the value of \(X\) that is most likely to occur.
Consequently, the mode is equal to the value of \(x\) at which the probability distribution function, \(P\begin{pmatrix}X = x \end{pmatrix}\), reaches a maximum.
A discrete random variable \(X\)can take-on the values: \[x = \left \{ 2, \ 3, \ 4, \ 5, \ 6 \right \}\] and has probability distribution function: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{x^2}{90}\]
The expected value of a discrete random variable \(X\) is the mean value (or average value) we could expect \(X\) to take if we were to repeat the experiment a large number of times. It is calculated with: \[E(X) = \sum x.P \begin{pmatrix} X = x \end{pmatrix} \] The expected value is also known as the mean \(\mu \) of the random variable, in which case we write: \[\mu = \sum x.P \begin{pmatrix} X = x \end{pmatrix}\] Note: it doesn't matter whether we refer to \(E(X)\) or \(\mu \), but it is important to know that they both refer to the same thing.
Consider the simple experiment of rolling a single unbiased dice once.
Define the discrete random variable \(X\) as:
\[X:\text{the value obtained when rolling the dice}\]
Find the mean value of this discrete random variable.
In the following tutorial we show how to find the mode and the mean of a discrete random variable, using the rules we just read (above).
We do this for a discrete random variable \(X\) that has the following probability distribution table:
Given a discrete random variable \(X\), we calculate its Variance, written \(Var\begin{pmatrix}X \end{pmatrix}\) or \(\sigma^2\), using one of the following two formula:
\[Var\begin{pmatrix}X \end{pmatrix} = \sum \begin{pmatrix}x - \mu \end{pmatrix}^2 . P\begin{pmatrix} X = x \end{pmatrix}\]
The standard deviation, \(\sigma\), of a discrete random variabe \(X\) tells us how far away from the mean \(\mu \) we can expect the value of \(X\) to be.
We calculate \(\sigma \) using the formula:
\[\sigma = \sqrt{ Var \begin{pmatrix} X \end{pmatrix}}\]
Note: it is important to realize and keep in mind that the value of \(\sigma \) is an average and could be expected to be observed after a sufficiently large number of trials.
A discrete random variable \(X\) can take the values \(x = \left \{ 3, \ 6, \ 7, \ 20 \right \}\) and has a probability distribution function \(P\begin{pmatrix} X = x \end{pmatrix} = \frac{x}{20}\).
In tutorial 2 we learn how to calculate the variance and standard deviation of a discrete random variable.
We do this for the following example:
A discrete random variable \(X\) can take the values \(x = \left \{ 1, \ 2, \ 3, \ 4 \right \}\).
It has probability distribution function \(P\begin{pmatrix}X = x \end{pmatrix} = \frac{x}{10}\).
Find: the variance and the standard deviation of \(X\).
Given a discrete random variable \(X\) and its cumulative distribution function \(P\begin{pmatrix} X \leq x \end{pmatrix} = F(x)\), the median of the discrete random variable \(X\) is the value \(M\) defined as: \[M = \frac{x_1+x_2}{2}\] Where:
The median \(M\) of \(X\) is the middle value.
The probability that \(X\) takes-on a value less than \(M\) is \(0.5\). Similarly, the probability that \(X\) takes-on a value greater than \(M\) is \(0.5\).
In other words there is an equal chance that \(X\) be greater or less than \(M\).
A discrete random variable \(X\) has the following cumulative probability distribution table:
In the following tutorial we learn how to find the median of a discrete random variable.
We start by reminding ourselves how to construct a cumulative probability distribution table and then learn how to use it to find the median value.
For this we suppose we're given a discrete random variable \(X\) with the following probability distribution table: