Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
then the maximun is
![**Mathematical Function Analysis**
Consider the given multivariable function:
\[ f(x, y) = e^{-3x^2 - 4y^2 + y} \]
In this function:
- \( f(x, y) \) represents the value of the function at any point \((x, y)\).
- The exponential term \[ e^{-3x^2 - 4y^2 + y} \] dictates the behavior and properties of the function.
### Explanation:
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
- The exponent \(-3x^2 - 4y^2 + y\) is a quadratic expression involving variables \(x\) and \(y\).
### Analysis:
- **Term -3x²**: Represents a quadratic decay in the x-direction which indicates that as \(x\) increases in magnitude (positive or negative), it contributes significantly to the negative exponent, causing the function \( f(x,y) \) to decrease.
- **Term -4y²**: Represents a quadratic decay in the y-direction, similar to the x-term. This quadratic term indicates that as \(y\) increases in magnitude (positive or negative), it also contributes significantly to decreasing the function \( f(x, y) \).
- **Term +y**: This linear term partially mitigates the effect of the \( -4y^2 \) for certain values of \(y\), enhancing the function's value along the positive y-direction.
### Conclusion:
The function \( f(x, y) = e^{-3x^2 - 4y^2 + y} \) will have a peak value where the negative exponential influence is minimized, which is centrally around \( x = 0 \) and a certain positive value of \( y \). Away from this central region, in both the x and y directions, the function value \( f(x, y) \) rapidly decreases towards zero due to the negative exponential terms.
Understanding this function is crucial for applications in fields such as statistical distribution, physics simulations, and higher-dimensional data analysis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F416064f9-c57b-46ee-9b50-14b257515654%2F153a9c1c-566a-4a71-a049-8047b322fe97%2Forly04i_processed.png&w=3840&q=75)
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