Method of Moments<\/span><\/h2>\n[\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row _builder_version=”4.16″ global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.16″ global_colors_info=”{}”][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
The Method of Moments is an inferential technique for estimating Population Parameters.<\/p>\n
These can be: the mean, variance, skewness, kurtosis and more.<\/p>\n
We call Moments to the Expected Values of a Random Variable.<\/p>\n
\u00a0We generate each moment by:<\/p>\n
. Arriving to the Moment Generating Function<\/p>\n
.Finding the corresponding derivative and plug in 0 where we have t<\/p>\n
To use this process we start with the Moment Generating Function.<\/p>\n
By definition:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn23.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
and at<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn24.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
p(x) refers to the discrete distribution’s PMF (Probability Mass Function)<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn22.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
f(x) refers to the continuous distribution’s PDF(Probability Density Function)<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn21.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
So the first step we check what type of distribution we are handling. If it is a discrete we use the summation function, if it is a continuous distribution, we use the integrating function.<\/p>\n
[\/et_pb_text][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
After arriving at the Moment Generating Function we can then work on each individual Moment.<\/p>\n
[\/et_pb_text][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
So each moment , is the same order number of the derivative of the Moment Generating Function with 0 plugged in. M1 is the first derivative, M2 the second derivative and so on.<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/M1-1.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/M2-1.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/m3-1.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/m4-1.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/mn-1.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
From here we can infer the following:<\/p>\n
M1 is the Expected Value<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/ex.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
And we can obtain the Variance of x by using M2 and M1<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn19.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
Moment Generating Function for the Bernoulli Distribution<\/strong><\/h3>\n[\/et_pb_text][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
The Bernoulli Distribution is discrete. It’s PMF is:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/Bernoulli-PMF.jpg” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
We start by the definition of the MGF, where:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn23.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
And since this is a discrete distribution , we use the discrete branch:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn22.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
We have x=0 and x=1, so we do the substitution:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn28.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
and get:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn30.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
which is our Moment Generating Function (MGF)<\/p>\n
[\/et_pb_text][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
So now, we can find M1 to get E[x] and with M2 we can then obtain Var(x). Let’s do that.<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn31.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
And to find Var(x) we need M2, so:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn32.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
Var(x) uses both moments, so we do the substitution and square M1 to arrive to the correct expression:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn19.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
So, the Variance is:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn33.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
Moment Generating Function for the Exponential Distribution<\/strong><\/h3>\n[\/et_pb_text][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
We start by firstly retrieving the Exponential distribution<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/poisson.png” _builder_version=”4.16″ min_height=”97px” global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
The exponential distribution is continuous so we use the integration branch.<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn21.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
we now substitute function branch in our MGF<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn35.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
to get our MGF:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn36.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
To find the first moment we do the first derivative of MGF and then plugin in 0 where t is<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn37.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
To find the second moment we do the second derivative of MGF and then plugin in 0 where t is<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn38.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
To find the variance we use the first and second moment.<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn19.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][et_pb_text _builder_version=”4.16″ global_colors_info=”{}”]<\/p>\n
So, the Variance is:<\/p>\n
[\/et_pb_text][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/CodeCogsEqn39.png” _builder_version=”4.16″ global_colors_info=”{}”][\/et_pb_image][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>","protected":false},"excerpt":{"rendered":"
In this article we will go over the method of moments, how to calculate the expected value and variance. Read more.<\/p>","protected":false},"author":1,"featured_media":8180,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[105],"tags":[430,421,459,107],"yoast_head":"\n
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