Probability and Statistics Lecture 12

Probability and Statistics Lecture 11

Info

Moment Generating Function (MGF) Calculation Prerequisites Moment Generating Functions

Questions

Question1

Find Moment Generating functiona and find first four moments.

Moment Generating Function (MGF) Calculation

The moment generating function (MGF) is given by:

$$M_X(t)=E(e^{tX})$$

Given the probability density function (PDF):

$$f(x)=\{\begin{array}{ll}2e^{-2x},& x\ge 0\\ 0,& \text{otherwise}\end{array}$$

We compute ( M_X(t) ) as:

$$M_X(t)={\int }_0^{\infty }{e}^{tx}f(x)dx$$

Substituting ( f(x) ):

$$M_X(t)={\int }_0^{\infty }{e}^{tx}\cdot 2e^{-2x}dx$$ $$=2{\int }_0^{\infty }{e}^{(t-2)x}dx$$

For convergence, we require ( t < 2 ). Evaluating the integral:

$$2{\int }_0^{\infty }{e}^{(t-2)x}dx=2{\left[\frac{e^{(t-2)x}}{t-2}\right]}_0^{\infty }$$ $$=2\left[\frac{e^{(t-2)\cdot \infty }-1}{t-2}\right]$$

Since ( e^{(t-2) \cdot \infty} \to 0 ) for ( t < 2 ):

$$M_X(t)=\frac{2}{2-t}=\frac{2}{1-t}$$

Finding the Moments

The moments are given by:

$${\mu }_n=E(X^n)=M_X^{(n)}(0)$$

First Moment (Mean ( $E(X)$ ))

Taking the first derivative:

$$M_X^{\prime }(t)=\frac{d}{dt}\left(\frac{2}{1-t}\right)$$

Using the derivative formula:

$$\frac{d}{dt}\left(\frac{1}{1-t}\right)=\frac{1}{(1-t{)}^2}$$

We get:

$$M_X^{\prime }(t)=2\cdot \frac{1}{(1-t{)}^2}=\frac{2}{(1-t{)}^2}$$

Evaluating at ( t = 0 ):

$$E(X)=M_X^{\prime }(0)=\frac{2}{(1-0{)}^2}=2$$

Second Moment ( $E(X^2)$ )

Taking the second derivative:

$$M_X^{\prime \prime }(t)=\frac{d}{dt}\left(\frac{2}{(1-t{)}^2}\right)$$

Using the chain rule:

$$M_X^{\prime \prime }(t)=2\cdot \frac{2}{(1-t{)}^3}=\frac{4}{(1-t{)}^3}$$

Evaluating at ( t = 0 ):

$$E(X^2)=M_X^{\prime \prime }(0)=\frac{4}{(1-0{)}^3}=4$$

Kurtosis and Skewness

Skewness

Skewness is the property of a Random Variable which gives us an idea about the shape of probability curve.
The coefficient depends on 3rd because

$$skewness=\frac{{\mu }_3}{{\mu }_2^{3/2}}$$

Depending on the value of coefficient of skewness

• $skewness>0⟹\text{The curve is positively skewed}$
• $skewness<0⟹\text{The curve is Negatively skewed}$

Kurtosis

Its the property of the Random Variable that gives us an idea about the flatness of the probability curve.

$${\beta }_2=\frac{{\mu }_4}{{\mu }_2^2}$$

Depending on the values

• ${\beta }_2=3⟹$ The curve is mesokurtic
• ${\beta }_2<3⟹$ The curve is platykurtic
• ${\beta }_2>3⟹$ The curve is leptokurtic

References

Information

• date: 2025.03.16
• time: 20:44

Link to original

References

Random Note snippets

Start with mean which is the first raw moment Then Variance which is the second central moment Then the third we use for skewness

Try watching 3Blue1Brown’s Convolution video to learn a cool facts about probability distribution.

Information

• date: 2025.02.05
• time: 14:06

Continued

Probability and Statistics Lecture 13