4. Assume X ~ N(u, o). Recall the moment generating function of X: M(t) = eut+io?. Also recall E(X*) = M (t)lt=0- dtk (a) Find the first moment E(X) (b) Find the second moment E(X²) (c) Find the third moment E(X3) (d) Find the fourth moment E(X4) (e) The skewness of X is defined by E[(-4) ]. Find the skewness of X. (Hint: use the binomial expansion). (f) The kurtosis of X is defined by El() ]. Find the kurtosis of X. (Hint: use the binomial expansion).
4. Assume X ~ N(u, o). Recall the moment generating function of X: M(t) = eut+io?. Also recall E(X*) = M (t)lt=0- dtk (a) Find the first moment E(X) (b) Find the second moment E(X²) (c) Find the third moment E(X3) (d) Find the fourth moment E(X4) (e) The skewness of X is defined by E[(-4) ]. Find the skewness of X. (Hint: use the binomial expansion). (f) The kurtosis of X is defined by El() ]. Find the kurtosis of X. (Hint: use the binomial expansion).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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I got the answers for a-d. I need help with e and f. THe answers to a-d are below:
A.) (Mu)
B.) (Mu)2 + (sigma)2
C.) 3(sigma)2(mu)+(mu)3
D.) (Mu)4+6(sigma)2(Mu)2+3(sigma)4
![### Problem 4: Moments and Properties of Normal Distribution
Assume \( X \sim N(\mu, \sigma) \). Recall the moment generating function of \( X \):
\[ M(t) = e^{\mu t + \frac{1}{2}t^2\sigma^2} \].
Also recall:
\[ E(X^k) = \frac{d^k}{dt^k} M(t) \bigg|_{t=0} \].
#### Tasks:
(a) **Find the first moment** \( E(X) \)
(b) **Find the second moment** \( E(X^2) \)
(c) **Find the third moment** \( E(X^3) \)
(d) **Find the fourth moment** \( E(X^4) \)
(e) **The skewness of \( X \)** is defined by:
\[ E\left( \left( \frac{X - \mu}{\sigma} \right)^3 \right) \].
*Find the skewness of \( X \). (Hint: use the binomial expansion.)*
(f) **The kurtosis of \( X \)** is defined by:
\[ E\left( \left( \frac{X - \mu}{\sigma} \right)^4 \right) \].
*Find the kurtosis of \( X \). (Hint: use the binomial expansion.)*
### Additional Notes
For parts (a) through (d), utilize the derivatives of the moment generating function \( M(t) \) evaluated at \( t=0 \).
For parts (e) and (f), consider the standardized variable \( Z = \frac{X - \mu}{\sigma} \) and apply the binomial theorem as a tool for expanding expressions when necessary. Skewness provides information about the asymmetry of the probability distribution of a real-valued random variable, while kurtosis gives insights about the tails and peak sharpness compared to a normal distribution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc91ff3af-9fac-40b4-8f24-485c7f78bb55%2F40a0d792-6c16-431d-aae1-fb3642e9c545%2Fwn17f8a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 4: Moments and Properties of Normal Distribution
Assume \( X \sim N(\mu, \sigma) \). Recall the moment generating function of \( X \):
\[ M(t) = e^{\mu t + \frac{1}{2}t^2\sigma^2} \].
Also recall:
\[ E(X^k) = \frac{d^k}{dt^k} M(t) \bigg|_{t=0} \].
#### Tasks:
(a) **Find the first moment** \( E(X) \)
(b) **Find the second moment** \( E(X^2) \)
(c) **Find the third moment** \( E(X^3) \)
(d) **Find the fourth moment** \( E(X^4) \)
(e) **The skewness of \( X \)** is defined by:
\[ E\left( \left( \frac{X - \mu}{\sigma} \right)^3 \right) \].
*Find the skewness of \( X \). (Hint: use the binomial expansion.)*
(f) **The kurtosis of \( X \)** is defined by:
\[ E\left( \left( \frac{X - \mu}{\sigma} \right)^4 \right) \].
*Find the kurtosis of \( X \). (Hint: use the binomial expansion.)*
### Additional Notes
For parts (a) through (d), utilize the derivatives of the moment generating function \( M(t) \) evaluated at \( t=0 \).
For parts (e) and (f), consider the standardized variable \( Z = \frac{X - \mu}{\sigma} \) and apply the binomial theorem as a tool for expanding expressions when necessary. Skewness provides information about the asymmetry of the probability distribution of a real-valued random variable, while kurtosis gives insights about the tails and peak sharpness compared to a normal distribution.
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