nyuuzyou/wikis huggingface dataset:
Moment Generating Function of Geometric Distribution/Formulation 1

== Theorem ==
Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p < 1$. 
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \dfrac {1 - p} {1 - p e^t}$
for $t < -\map \ln p$, and is undefined otherwise.
== Proof ==
From the definition of the geometric distribution, $X$ has probability mass function:
:$\map \Pr {X = k} = \paren {1 - p} p^k$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^\infty \map \Pr {X = k} e^{k t}$
So:
By Sum of Infinite Geometric Sequence, for this sum to be convergent we must have: 
:$\size {p e^t} < 1$
In the case $p = 0$, this demand is satisfied immediately regardless of $t$.
Otherwise, as both $e^t$ and $p$ are positive: 
:$e^t < \dfrac 1 p$
So, by Logarithm of Power: 
:$t < -\map \ln p$
is the range of $t$ for which the expectation is well-defined.
Now applying Sum of Infinite Geometric Sequence, we have for this range of $t$: 
:$\ds \map {M_X} t = \paren {1 - p} \sum_{k \mathop = 0}^\infty \paren {p e^t}^k = \frac {1 - p} {1 - p e^t}$
== Examples ==
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 1/Examples

== Examples of Use of Moment Generating Function of Geometric Distribution ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
=== First Moment ===
=== Second Moment ===
=== Third Moment ===
=== Fourth Moment ===
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 1/Examples/First Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 1 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
The first moment generating function of $X$ is given by: 
:$\map { {M_X}'} t = \dfrac {\paren {1 - p} p e^t} {\paren {1 - p e^t}^2}$
== Proof ==
We have:
 
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 1/Examples/Fourth Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 1 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
The fourth moment generating function of $X$ is given by: 
:$\map { {M_X}^{\paren 4} } t = p \paren {1 - p } e^t \paren {\dfrac {1 + 11p e^t + 11 p^2 e^{2t} + p^3 e^{3t} } {\paren {1 - p e^t}^5 } }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 1/Examples/Second Moment

== Examples of Use of Moment Generating Function of Geometric Distribution ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
The second moment generating function of $X$ is given by: 
:$\map { {M_X}} t = p \paren {1 - p} e^t \paren {\dfrac {1 + p e^t} {\paren {1 - p e^t}^3} }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 1/Examples/Third Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 1 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = \paren {1 - p} p^k$
The third moment generating function of $X$ is given by: 
:$\map { {M_X}} t = p \paren {1 - p } e^t \paren {\dfrac {1 + 4p e^t + p^2 e^{2t} } {\paren {1 - p e^t}^4 } }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2

== Theorem ==
Let $X$ be a discrete random variable with a geometric distribution with parameter $p$ for some $0 < p < 1$. 
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \dfrac p {1 - \paren {1 - p} e^t}$
for $t < -\map \ln {1 - p}$, and is undefined otherwise.
== Proof ==
From the definition of the geometric distribution, $X$ has probability mass function:
:$\map \Pr {X = k} = p \paren {1 - p}^k$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \sum_{k \mathop = 0}^\infty \map \Pr {X = k} e^{k t}$
So:
By Sum of Infinite Geometric Sequence, for this sum to be convergent we must have: 
:$\size {\paren {1 - p} e^t} < 1$
In the case $p = 1$, this demand is satisfied immediately regardless of $t$.
Otherwise, as both $e^t$ and $p$ are positive: 
:$e^t < \dfrac 1 {1 - p}$
So, by Logarithm of Power: 
:$t < -\map \ln {1 - p}$
is the range of $t$ for which the expectation is well-defined.
Now applying Sum of Infinite Geometric Sequence, we have for this range of $t$: 
:$\ds \map {M_X} t = p \sum_{k \mathop = 0}^\infty \paren {\paren {1 - p} e^t}^k = \frac p {1 - \paren {1 - p } e^t}$
=== Examples ===
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2/Examples

== Examples of Use of Moment Generating Function of Geometric Distribution ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
=== First Moment ===
=== Second Moment ===
=== Third Moment ===
=== Fourth Moment ===
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2/Examples/First Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 2 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
The first moment generating function of $X$ is given by: 
:$\map { {M_X}'} t = \dfrac {p \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^2 }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2/Examples/Fourth Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 2 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
The fourth moment generating function of $X$ is given by: 
:$\map { {M_X}^{\paren 4} } t = p \paren {1 - p } e^t \paren {\dfrac {1 + 11 \paren {1 - p} e^t + 11 \paren {1 - p}^2 e^{2t} + \paren {1 - p}^3 e^{3t} } {\paren {1 - \paren {1 - p} e^t}^5 } }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2/Examples/Second Moment

== Examples of Use of Moment Generating Function of Geometric Distribution ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
The second moment generating function of $X$ is given by: 
:$\map { {M_X}} t = p \paren {1 - p} e^t \paren {\dfrac {1 + \paren {1 - p} e^t } {\paren {1 - \paren {1 - p} e^t}^3 } }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Geometric Distribution/Formulation 2/Examples/Third Moment

== Examples of Use of Moment Generating Function of Geometric Distribution/Formulation 2 ==
Let $X \sim \Geometric p$ for some $0 < p < 1$, where $\Geometric p$ is the Geometric distribution.
:$\map X \Omega = \set {0, 1, 2, \ldots} = \N$
:$\map \Pr {X = k} = p \paren {1 - p}^k$
The third moment generating function of $X$ is given by: 
:$\map { {M_X}} t = p \paren {1 - p} e^t \paren {\dfrac {1 + 4 \paren {1 - p} e^t + \paren {1 - p}^2 e^{2t} } {\paren {1 - \paren {1 - p} e^t}^4 } }$
== Proof ==
We have:
Category:Moment Generating Function of Geometric Distribution
Moment Generating Function of Linear Combination of Independent Random Variables

== Theorem ==
Let $X_1, X_2, \ldots, X_n$ be independent random variables. 
Let $k_1, k_2, \ldots, k_n$ be real numbers.
Let: 
:$\ds X = \sum_{i \mathop = 1}^n k_i X_i$
Let $M_{X_i}$ be the moment generating function of $X_i$ for $1 \le i \le n$.
Then: 
:$\ds \map {M_X} t = \prod_{i \mathop = 1}^n \map {M_{X_i}} {k_i t}$
for all $t$ such that $M_{X_i}$ exists for all $1 \le i \le n$.
== Proof == 
Category:Moment Generating Functions
Moment Generating Function of Linear Transformation of Random Variable

== Theorem ==
Let $X$ be a random variable. 
Let $\alpha$ and $\beta$ be real numbers.
Let $Z = \alpha X + \beta$.
Let $M_X$ be the moment generating function of $X$. 
Then the moment generating function of $Z$, $M_Z$, is given by: 
:$\map {M_Z} t = e^{\beta t} \map {M_X} {\alpha t}$
== Proof ==
Category:Moment Generating Functions
Moment Generating Function of Logistic Distribution

== Theorem ==
Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \begin {cases} \map \exp {\mu t} \map \Beta {\paren {1 - s t}, \paren {1 + s t} } & \size t < \dfrac 1 s \\ \text {does not exist} & \size t \ge \dfrac 1 s \end {cases}$
where $\Beta$ denotes the beta function
== Proof == 
From the definition of the logistic distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {\map \exp {-\dfrac {\paren {x - \mu} } s} } {s \paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$
So: 
:$\ds \map {M_X} t = \frac 1 s \int_{-\infty}^\infty \dfrac {\map \exp {t x} \map \exp {-\dfrac {\paren {x - \mu} } s} } {\paren {1 + \map \exp {-\dfrac {\paren {x - \mu} } s} }^2} \rd x$
let:
and also:
Then:
Note that by definition of the beta function, $\Beta: \C \times \C \to \C$ is defined only for $\map \Re x, \map \Re y > 0$:
Therefore:
:$1 - s t > 0 \leadsto t < \dfrac 1 s$
and:
:$1 + s t > 0 \leadsto t > -\dfrac 1 s$
Therefore, the moment generating function is defined only when $\size t < \dfrac 1 s$.
== Examples ==
Category:Moment Generating Functions
Category:Logistic Distribution
Category:Moment Generating Function of Logistic Distribution
Moment Generating Function of Logistic Distribution/Examples

== Examples of Use of Moment Generating Function of Logistic Distribution ==
Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Let $\size t < \dfrac 1 s$.
=== First Moment ===
=== Second Moment ===
=== Third Moment ===
=== Fourth Moment ===
=== Derivatives of Moment Generating Function of Logistic Distribution‎ ===
Category:Moment Generating Function of Logistic Distribution
Moment Generating Function of Logistic Distribution/Examples/First Moment

== Examples of Use of Moment Generating Function of Logistic Distribution ==
Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Let $\size t < \dfrac 1 s$.
The first moment generating function of $X$ is given by: 
:$\ds \map { {M_X}'} t = \map \exp {\mu t} \paren {\mu \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u - s \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u}$
== Proof ==
We have:
Category:Moment Generating Function of Logistic Distribution
Moment Generating Function of Logistic Distribution/Examples/Second Moment

== Examples of Use of Moment Generating Function of Logistic Distribution ==
Let $X$ be a continuous random variable which satisfies the logistic distribution:
:$X \sim \map {\operatorname {Logistic} } {\mu, s}$
for some $\mu \in \R, s \in \R_{> 0}$.
Let $\size t < \dfrac 1 s$.
The second moment generating function of $X$ is given by: 
:$\ds \map { {M_X}} t = \map \exp {\mu t} \paren {\mu^2 \int_{\to 0}^{\to 1} \paren {\dfrac {1 - u} u}^{-s t} \rd u - 2 s \mu \int_{\to 0}^{\to 1} \map \ln {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u + s^2 \int_{\to 0}^{\to 1} \map {\ln^2} {\dfrac {1 - u} u} \paren {\dfrac {1 - u} u}^{-s t} \rd u}$
== Proof ==
We have:
Category:Moment Generating Function of Logistic Distribution
Moment Generating Function of Normal Distribution

== Theorem ==
Let $X \sim \Gaussian \mu {\sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
== Proof == 
From the definition of the normal distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac 1 {\sigma \sqrt{2 \pi} } \map \exp {-\dfrac {\paren {x - \mu}^2} {2 \sigma^2} }$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_{-\infty}^\infty e^{t x} \map {f_X} x \rd x$
So: 
== Examples ==
== Sources ==
* : Appendix $13$: Probability distributions
* : Appendix $15$: Probability distributions
Category:Moment Generating Function of Normal Distribution
Category:Moment Generating Functions
Category:Normal Distribution
Moment Generating Function of Normal Distribution/Examples

== Examples of Use of Moment Generating Function of Normal Distribution ==
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
=== First Moment ===
=== Second Moment ===
=== Third Moment ===
=== Fourth Moment ===
Category:Moment Generating Function of Normal Distribution
Moment Generating Function of Normal Distribution/Examples/First Moment

== Examples of Use of Moment Generating Function of Normal Distribution ==
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
The first moment generating function of $X$ is given by: 
:$\map { {M_X}'} t = \paren {\mu + \sigma^2 t} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
== Proof ==
We have: 
Category:Moment Generating Function of Normal Distribution
Moment Generating Function of Normal Distribution/Examples/Fourth Moment

== Examples of Use of Moment Generating Function of Normal Distribution ==
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
The fourth moment generating function of $X$ is given by: 
:$\map { {M_X}^{\paren 4} } t = \paren {3 \sigma^4 + 6 \sigma^2 \paren {\mu + \sigma^2 t}^2 + \paren {\mu + \sigma^2 t}^4} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
== Proof ==
We have: 
Category:Moment Generating Function of Normal Distribution
Moment Generating Function of Normal Distribution/Examples/Second Moment

== Examples of Use of Moment Generating Function of Normal Distribution ==
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
The second moment generating function of $X$ is given by: 
:$\map { {M_X}' '} t = \paren {\sigma^2 + \paren {\mu + \sigma^2 t}^2 } \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
== Proof ==
We have: 
Category:Moment Generating Function of Normal Distribution
Moment Generating Function of Normal Distribution/Examples/Third Moment

== Examples of Use of Moment Generating Function of Normal Distribution ==
Let $X \sim N \paren {\mu, \sigma^2}$ for some $\mu \in \R, \sigma \in \R_{> 0}$, where $N$ is the normal distribution.
The third moment generating function of $X$ is given by: 
:$\map { {M_X}' ' '} t = \paren {3 \sigma^2 \paren {\mu + \sigma^2 t} + \paren {\mu + \sigma^2 t}^3} \map \exp {\mu t + \dfrac 1 2 \sigma^2 t^2}$
== Proof ==
We have: 
Category:Moment Generating Function of Normal Distribution
Moment Generating Function of Pareto Distribution

== Theorem ==
Let $X$ be a continuous random variable with a Pareto distribution with parameters a and b for $a, b \in \R_{> 0}$. 
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = \begin {cases} a \paren {-b t}^a \map \Gamma {-a, -b t} & t < 0 \\ 1 & t = 0 \\ \text {does not exist} & t > 0 \end {cases}$
== Proof == 
From the definition of the Pareto distribution, $X$ has probability density function:
:$\map {f_X} x = \dfrac {a b^a} {x^{a + 1} }$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect { e^{t X} } = \int_b^\infty e^{t x} \map {f_X} x \rd x$
First take $t < 0$. 
Then:
:$\ds \map {M_X} t = a b^a \int_b^\infty x^{-\paren {a + 1} } e^{t x} \rd x $
let:
Then: 
Now take $t = 0$. 
Our integral becomes: 
Finally take $t > 0$.
Then:
As a consequence of Exponential Dominates Polynomial, we have:
:$x^a < e^{t x}$
for sufficiently large $x$. 
Therefore, in this case, the integrand increases without bound. 
We conclude that the integral is divergent.
Hence $\expect {e^{t X} }$ does not exist for $t > 0$. 
Category:Moment Generating Functions
Category:Pareto Distribution
Moment Generating Function of Poisson Distribution

== Theorem ==
Let $X$ be a discrete random variable with a Poisson distribution with parameter $\lambda$ for some $\lambda \in \R_{> 0}$.
Then the moment generating function $M_X$ of $X$ is given by: 
:$\map {M_X} t = e^{\lambda \paren {e^t - 1} }$
== Proof ==
From the definition of the Poisson distribution, $X$ has probability mass function:
:$\map \Pr {X = n} = \dfrac {\lambda^n e^{-\lambda} } {n!}$
From the definition of a moment generating function:
:$\ds \map {M_X} t = \expect {e^{t X} } = \sum_{n \mathop = 0}^\infty \map \Pr {X = n} e^{t n}$
So: 
== Sources ==
* : Appendix $13$: Probability distributions
* : Appendix $15$: Probability distributions
Category:Moment Generating Functions
Category:Poisson Distribution
Moment in terms of Moment Generating Function

== Theorem == 
Let $X$ be a random variable. 
Let $M_X$ be the moment generating function of $X$.
Then: 
:$\expect {X^n} = \map { {M_X}^{\paren n} } 0$
where:
:$n$ is a non-negative integer
:${M_X}^{\paren n}$ denotes the $n$th derivative of $M_X$
:$\expect {X^n}$ denotes the expectation of $X^n$.
== Proof == 
Setting $t = 0$ yields the result. 
== Sources ==
* 
* 
Category:Moment Generating Functions
Category:Moments (Probability Theory)
Moment of Inertia of Uniform Circular Lamina through Midpoint about Perpendicular Axis

== Theorem ==
Let $\LL$ be a uniform lamina of mass $M$ in the shape of a circle whose radius is $a$.
Let $\AA$ be the straight line through the centroid of $\LL$ perpendicular to $\LL$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M a^2} 2$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Uniform Laminae
Category:Circles
Moment of Inertia of Uniform Ellipsoid about Axis of Ellipsoid

== Theorem ==
Let $\EE$ be a solid ellipsoid of uniform density of mass $M$ whose axes are of length $2 a$, $2 b$ and $2 c$.
Let $\AA$ be the straight line coinciding with the axis of $\EE$ of length $2 c$.
Then the moment of inertia $\II$ of $\EE$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 5$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Ellipsoids
Moment of Inertia of Uniform Ellipsoidal Shell about Axis of Ellipsoid

== Theorem ==
Let $\SS$ be a uniform lamina of mass $M$ in the shape of the surface of an ellipsoid whose axes are of length $2 a$, $2 b$ and $2 c$.
Let $\AA$ be the straight line coincident with the axis of $\EE$ of length $2 c$.
Then the moment of inertia $\II$ of $\SS$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 3$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Uniform Laminae
Category:Ellipsoids
Moment of Inertia of Uniform Elliptical Lamina about Axis of Ellipse

== Theorem ==
Let $\LL$ be a uniform lamina of mass $M$ in the shape of a ellipse whose axes are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line coinciding with the axis of $\LL$ of length $2 a$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M b^2} 4$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Uniform Laminae
Category:Ellipses
Moment of Inertia of Uniform Elliptical Lamina through Midpoint about Perpendicular Axis

== Theorem ==
Let $\LL$ be a uniform lamina of mass $M$ in the shape of a ellipse whose axes are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line through the centroid of $\LL$ perpendicular to $\LL$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 4$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Uniform Laminae
Category:Ellipses
Moment of Inertia of Uniform Rectangular Lamina through Midpoint about Perpendicular Axis

== Theorem ==
Let $\LL$ be a uniform lamina of mass $M$ in the shape of a rectangle whose sides are of length $2 a$ and $2 b$.
Let $\AA$ be the straight line through the centroid of $\LL$ perpendicular to $\LL$.
Then the moment of inertia $\II$ of $\LL$ about $\AA$ is given by:
:$\II = \dfrac {M \paren {a^2 + b^2} } 3$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Uniform Laminae
Category:Rectangles
Moment of Inertia of Uniform Rod through Midpoint

== Theorem ==
Let $\RR$ be a uniform rod of length $2 a$ and mass $M$.
Let $\AA$ be the straight line through the midpoint of $\RR$ perpendicular to $\RR$.
Then the moment of inertia $\II$ of $\RR$ about $\AA$ is given by:
:$\II = \dfrac {M a^2} 3$
== Proof ==
== Sources ==
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
* : Appendix: Table $5$ Moments of inertia Values for certain uniform bodies of mass $M$ about certain axes.
Category:Moments of Inertia
Category:Rods