How to Expand in Statistics

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Expanding in statistics will require you to expand on certain expressions in order to form a calculation. Expand in statistics with help from an experienced mathematics professional in this free video clip.

Part of the Video Series: Trigonometry, Graphs, & Other Math Tips
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Hello, my name is Walter Unglaub, and this is how to expand in statistics. So in statistics and probability, much like in physics or other sciences, there are many mathematical tools and theorems that can be used to either simplify or expand certain expressions in order to form a calculation. For this problem and this example, we're going to consider the binomial theorem which states that if you have some polynomial here, some x plus y raised to the power n, it can be written as a sum of k going from zero to n, which is the power, of a multiplicity factor times x raised to k, times y raised to n minus k. So, this multiplicity factor, I should denote as a function of n and k, and it's equal n factorial divided by k factorial time n minus k factorial. This is very widely used in many fields, and will be integral to the binomial theorem. So if I consider as an example x plus one raised to the power two, I can actually calculate what this is using the binomial theorem, and then I'm going to directly calculate this and check to make sure that the theorem holds. So, what I have is n is equal to two, so my sum is gonna go from k is equal to zero to two, and in my multiplicity I have two and then k, then I'm going to have x raised to the power k, and y you'll notice is equal to one. So if I have one raised to n minus k, then this term will always be equal to one, because the base is unity. So the sum is going to be equal to two factorial over zero factorial, two minus zero factorial, times x to the zero, plus two factorial over one factorial, two minus one factorial times x raised to the power of one, plus two factorial over two factorial, and then two minus two factorial, times x raised to the power of two. Zero factorial of course is equal to one, so I'm gonna have two factorials over two factorial, and this is equal to one. X raised to the zero power is equal to one. So for my first term I get one. For my second term I have two factorial up top and one time one is equal to one. So I have two time x. And then finally for my third term I have two factorial divided two factorial, which is, which is just one, and zero factorial down here which is also equal to one. So I end up with x-squared. And we see that this is indeed what x plus one-squared is equal to. It's equal to x plus one, times x plus one, and this product turns out to be x-squared plus 2x plus one. My name is Walter Unglaub, and this is how to expand in statistics.

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