Chi-Squared Distribution

Welcome back, folks! This is going to be a short lecture where
we introduce to you the Chi-squared Distribution. For starters, we define a denote a Chi-Squared
distribution with the capital Greek letter Chi, squared followed by a parameter “k”
depicting the degrees of freedom. Therefore, we read the following as “Variable
“Y” follows a Chi-Square distribution with 3 degrees of freedom”. Alright! Let’s get started! Very few events in real life follow such a
distribution. In fact, Chi-Squared is mostly featured in
statistical analysis when doing hypothesis testing and computing confidence intervals. In particular, we most commonly find it when
determining the goodness of fit of categorical values. That is why any example we can give you would
feel extremely convoluted to anyone not familiar with statistics. Alright! Now, let’s explore the graph of the Chi-Squared
distribution. Just by looking at it, you can tell the distribution
is not symmetric, but rather – asymmetric. Its graph is highly-skewed to the right. Furthermore, the values depicted on the X-axis
start form 0, rather than some negative number. This, by the way, shows you yet another transformation. Elevating the Student’s T distribution to
the second power gives us the Chi-squared and vice versa: finding the square root of
the Chi-squared distribution gives us the Student’s T. Great! So, a convenient feature of the Chi-Squared
distribution is that it also contains a table of known values, just like the Normal or Students’–T
distributions. The expected value for any Chi-squared distribution
is equal to its associated degrees of freedom, k. Its variance is equal to two times the degrees
of freedom, or simply 2 times k.

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