Physical or online
by wimpykid, Apr 30, 2025, 6:49 AM
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Generating Functions
by greenplanet2050, Apr 29, 2025, 10:42 PM
So im learning generating functions and i dont really understand why 
can someone help
thank you
can someone help
thank you
Transformation of a cross product when multiplied by matrix A
by Math-lover1, Apr 29, 2025, 10:29 PM
I was working through AoPS Volume 2 and this statement from Chapter 11: Cross Products/Determinants confused me.
I understand that this is true for the unit vectors 
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.
How do I prove this statement easily with any two 2D vectors?
AoPS Volume 2 wrote:
A quick comparison of 
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.
to the cross product 
reveals that a negative determinant [of 
] corresponds to a matrix which reverses the direction of the cross product of two vectors.
and 
, but am confused on how to prove this statement for general vectors 
and 
although its supposed to be a quick comparison.How do I prove this statement easily with any two 2D vectors?
This post has been edited 1 time. Last edited by Math-lover1, Yesterday at 10:30 PM
Reason: edit
Reason: edit
.
.Inequalities
by sqing, Apr 25, 2025, 9:19 AM
Let ![$ a,b \in [0 ,1] . $](//latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
This post has been edited 4 times. Last edited by sqing, Apr 25, 2025, 9:53 AM
Geometric inequality
by ReticulatedPython, Apr 22, 2025, 5:12 PM
Let 
and 
be points on a plane such that 
, where 
is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
and 
be points on a plane such that 
, where 
is a positive integer. Let 
be the set of all points 
such that 
, where 
is a real number. The path that traces is continuous, and the value of is minimized. Prove that is rational for all positive integers 
This post has been edited 2 times. Last edited by ReticulatedPython, Apr 22, 2025, 8:06 PM
Three variables inequality
by Headhunter, Apr 20, 2025, 6:58 AM
,
,
Prove that at least one of
, 
, 
is not greater than 
.I assume that all are greater than it, but can't go more.
The Bell Curve
by aoum, Apr 14, 2025, 11:37 PM
The Bell Curve: Mathematics of the Normal Distribution
The Bell Curve is the common name for the graph of the normal distribution, one of the most fundamental and widely used probability distributions in mathematics, statistics, and the sciences. Its characteristic shape — symmetric, centered, and tapering off — resembles a bell.
1. Definition
The normal distribution with mean
and standard deviation 
has the probability density function (PDF):
This function satisfies:
The standard normal distribution is the special case when
and 
:
2. Properties
3. Empirical Rule (68-95-99.7 Rule)
In a normal distribution:
This makes the bell curve a natural model for random variation in measurements.
4. Cumulative Distribution Function (CDF)
The cumulative distribution function is:
There is no closed-form expression in terms of elementary functions, but it is closely related to the error function:
![$$
\Phi(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right) \right]
$$](//latex.artofproblemsolving.com/b/e/9/be940b9663675e1d94a3d46a4893045eaf910760.png)
5. Central Limit Theorem (CLT)
The Central Limit Theorem states that the sum (or average) of many independent random variables, regardless of their individual distributions, tends to a normal distribution as the number of variables grows.
This explains why the bell curve appears so frequently in nature, science, and statistics.
6. Standardization
Any normal variable
can be converted to a standard normal variable 
using:
This transformation allows use of standard normal tables.
7. Applications
The bell curve is used to model:
8. Moments and Moment Generating Function
The moment generating function (MGF) of is:
![$$
M_X(t) = \mathbb{E}[e^{tX}] = \exp\left( \mu t + \frac{1}{2} \sigma^2 t^2 \right)
$$](//latex.artofproblemsolving.com/5/6/2/562077a258eae6445c4da40820a07558d5eff915.png)
The-th moment is:
![$$
\mathbb{E}[X^n] = M_X^{(n)}(0)
$$](//latex.artofproblemsolving.com/4/8/c/48c7ba5e4126111f3a60173a597a665fc143125b.png)
9. Derivation from First Principles (Sketch)
One way to derive the normal distribution is from the requirement that it maximizes entropy among all continuous distributions with a given mean and variance. Another is via the De Moivre–Laplace limit theorem (a special case of the CLT).
Alternatively, it arises as the limit of binomial distributions:
10. Reference Integrals
The key integral (used in normalization) is:
and
11. References
The Bell Curve is the common name for the graph of the normal distribution, one of the most fundamental and widely used probability distributions in mathematics, statistics, and the sciences. Its characteristic shape — symmetric, centered, and tapering off — resembles a bell.
Probability density function
The red curve is the standard normal distribution.
Cumulative distribution function
1. Definition
The normal distribution with mean
and standard deviation 
has the probability density function (PDF):
This function satisfies:
- It is symmetric about
- It is maximized at
- The area under the curve is 1:
The standard normal distribution is the special case when
and 
:
2. Properties
- Mean:
- Variance:
- Standard deviation:
- Mode and median: both equal to
- The curve is bell-shaped and asymptotic to the
-axis.
-3. Empirical Rule (68-95-99.7 Rule)
In a normal distribution:
- About 68.27% of the data falls within one standard deviation of the mean:
![$[\mu - \sigma, \mu + \sigma]$](//latex.artofproblemsolving.com/4/c/d/4cd77641d31296f8f62cd8649328717c9de5f5e0.png)
- About 95.45% lies within two standard deviations
- About 99.73% lies within three standard deviations
![$[\mu - \sigma, \mu + \sigma]$](http://latex.artofproblemsolving.com/4/c/d/4cd77641d31296f8f62cd8649328717c9de5f5e0.png)
This makes the bell curve a natural model for random variation in measurements.
4. Cumulative Distribution Function (CDF)
The cumulative distribution function is:
There is no closed-form expression in terms of elementary functions, but it is closely related to the error function:
![$$
\Phi(x) = \frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x}{\sqrt{2}} \right) \right]
$$](http://latex.artofproblemsolving.com/b/e/9/be940b9663675e1d94a3d46a4893045eaf910760.png)
5. Central Limit Theorem (CLT)
The Central Limit Theorem states that the sum (or average) of many independent random variables, regardless of their individual distributions, tends to a normal distribution as the number of variables grows.
This explains why the bell curve appears so frequently in nature, science, and statistics.
6. Standardization
Any normal variable
can be converted to a standard normal variable 
using:
This transformation allows use of standard normal tables.
7. Applications
The bell curve is used to model:
- Measurement errors
- Heights and weights
- IQ scores
- Test scores
- Brownian motion
- Many statistical inference procedures (confidence intervals, hypothesis testing)
8. Moments and Moment Generating Function
The moment generating function (MGF) of
is:![$$
M_X(t) = \mathbb{E}[e^{tX}] = \exp\left( \mu t + \frac{1}{2} \sigma^2 t^2 \right)
$$](http://latex.artofproblemsolving.com/5/6/2/562077a258eae6445c4da40820a07558d5eff915.png)
The
-th moment is:![$$
\mathbb{E}[X^n] = M_X^{(n)}(0)
$$](http://latex.artofproblemsolving.com/4/8/c/48c7ba5e4126111f3a60173a597a665fc143125b.png)
9. Derivation from First Principles (Sketch)
One way to derive the normal distribution is from the requirement that it maximizes entropy among all continuous distributions with a given mean and variance. Another is via the De Moivre–Laplace limit theorem (a special case of the CLT).
Alternatively, it arises as the limit of binomial distributions:
10. Reference Integrals
The key integral (used in normalization) is:
and
11. References
- Wikipedia: Normal Distribution
- Feller, W. An Introduction to Probability Theory and Its Applications
- AoPS Wiki: Normal Distribution
- Papoulis, A. Probability, Random Variables, and Stochastic Processes
April 2025
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