variance

[logistic_guy]

A random variable \(\displaystyle X\) that assumes the values \(\displaystyle x_1, x_2,\cdots, x_k\) is called a discrete uniform random variable if its probability mass function is [imath]f(x) = \frac{1}{k}[/imath] for all of \(\displaystyle x_1, x_2,\cdots, x_k\) and \(\displaystyle 0\) otherwise. Find the mean and variance of \(\displaystyle X\).

 

[khansaheb]

A random variable \(\displaystyle X\) that assumes the values \(\displaystyle x_1, x_2,\cdots, x_k\) is called a discrete uniform random variable if its probability mass function is [imath]f(x) = \frac{1}{k}[/imath] for all of \(\displaystyle x_1, x_2,\cdots, x_k\) and \(\displaystyle 0\) otherwise. Find the mean and variance of \(\displaystyle X\).

Please define random variable.

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[logistic_guy]

Find the mean

\(\displaystyle \textcolor{indigo}{\bold{mean}}\rightarrow \mu\)

So, for a random variable \(\displaystyle X\), we have:

\(\displaystyle \mu = E(X) = \sum_{i = 1}^{k}x_if(x_i) = \sum_{i = 1}^{k}x_i\frac{1}{k} = \textcolor{blue}{\frac{1}{k}\sum_{i = 1}^{k}x_i}\)

 

[logistic_guy]

Find the variance

\(\displaystyle \textcolor{indigo}{\bold{variance}}\rightarrow \sigma^2\)

So, for a random variable \(\displaystyle X\), we have:

\(\displaystyle \sigma^2 = E(X^2) - \mu^2 = \sum_{i = 1}^{k}x^2_if(x_i) - \left(\sum_{i = 1}^{k}x_if(x_i)\right)^2\)

Or

\(\displaystyle \sigma^2 = \frac{1}{k}\sum_{i = 1}^{k}x^2_i - \left(\frac{1}{k}\sum_{i = 1}^{k}x_i\right)^2\)

Or if you wanna be \(\displaystyle \textcolor{green}{\bold{FANCY.}}\)

\(\displaystyle \sigma^2 = \frac{1}{k}\sum_{i = 1}^{k}(x_i - \mu)^2\)

\(\displaystyle \textcolor{red}{\huge\bold{How?}}\)