{"id":7814,"date":"2022-09-22T22:50:27","date_gmt":"2022-09-22T22:50:27","guid":{"rendered":"http:\/\/euseguros.pt\/?page_id=7814"},"modified":"2022-09-25T17:26:02","modified_gmt":"2022-09-25T17:26:02","slug":"method-of-moments","status":"publish","type":"page","link":"https:\/\/euseguros.pt\/en\/statistics-full-course\/method-of-moments\/","title":{"rendered":"Method Of Moments"},"content":{"rendered":"

[et_pb_section fb_built=”1″ fullwidth=”on” _builder_version=”4.4.3″][et_pb_fullwidth_menu menu_id=”410″ menu_style=”inline_centered_logo” fullwidth_menu=”on” _builder_version=”4.4.3″ menu_font_size=”20px” custom_margin=”|||0px|false|false”][\/et_pb_fullwidth_menu][\/et_pb_section][et_pb_section fb_built=”1″ _builder_version=”4.4.3″][et_pb_row _builder_version=”4.4.3″][et_pb_column type=”4_4″ _builder_version=”4.4.3″][et_pb_text _builder_version=”4.4.3″]<\/p>\n

Method of Moments<\/span><\/h2>\n

[\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row _builder_version=”4.4.3″][et_pb_column type=”4_4″ _builder_version=”4.4.3″][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″]<\/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=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/M2-1.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/m3-1.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/m4-1.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/mn-1.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/p>\n

Moment Generating Function for the Bernoulli Distribution<\/strong><\/h3>\n

[\/et_pb_text][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/p>\n

which is our Moment Generating Function (MGF)<\/p>\n

[\/et_pb_text][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/p>\n

Moment Generating Function for the Exponential Distribution<\/strong><\/h3>\n

[\/et_pb_text][et_pb_text _builder_version=”4.4.3″ hover_enabled=”0″]<\/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=”4.4.3″ min_height=”97px”][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][et_pb_image src=”http:\/\/euseguros.pt\/wp-content\/uploads\/2022\/09\/varx-2.png” _builder_version=”4.4.3″][\/et_pb_image][et_pb_text _builder_version=”4.4.3″]<\/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=”4.4.3″][\/et_pb_image][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>","protected":false},"excerpt":{"rendered":"

Method of Moments. In statistics, the method of moments is a method of estimation of population parameters. Learn more about this method.<\/p>","protected":false},"author":1,"featured_media":0,"parent":7674,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"yoast_head":"\nMethod Of Moments - EU Seguros<\/title>\n<meta name=\"description\" content=\"Method of Moments. In statistics, the method of moments is a method of estimation of population parameters. 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