Derivation of Exponential Equation

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    Derivation
In summary, the given expression simplifies to an exponential term with a coefficient of n times lambda divided by mu, multiplied by a product of two exponential terms with coefficients of negative lambda divided by 2 times mu squared and negative lambda divided by 2, respectively. The denominator term is 2 times mu squared.
  • #1
roadworx
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hi,

I have:

[tex]exp \left(-\frac{\lambda}{2^\mu^2} \sum_{i=1}^n \frac{(x_i-\mu)^2}{x_i}\right)[/tex]

I am trying to work out why this simplifies to:

[tex]exp\left(\frac{n \lambda}{\mu}\right) exp \left( -\frac{\lambda}{2^\mu^2} \sum_{i=1}^n x_i -\frac{ \lambda}{2} \sum_{i=1}^n \frac{1}{x_i} \right) [/tex]

Any ideas?
 
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  • #2
roadworx said:
hi,

I have:

[tex]exp \left(-\frac{\lambda}{2^\mu^2} \sum_{i=1}^n \frac{(x_i-\mu)^2}{x_i}\right)[/tex]

I am trying to work out why this simplifies to:

[tex]exp\left(\frac{n \lambda}{\mu}\right) exp \left( -\frac{\lambda}{2^\mu^2} \sum_{i=1}^n x_i -\frac{ \lambda}{2} \sum_{i=1}^n \frac{1}{x_i} \right) [/tex]

Any ideas?
Multiply out that square: [itex](x_i- \mu)^2= x_i^2- 2\mu x_i+ \mu^2[/itex]. Dividing by [itex]x_i[/itex] gives
[tex]\frac{(x_i- \mu)^2}{x_i}= x_i- 2\mu+ \frac{mu^2}{x_i}[/tex]
so that sum can be written as three different sums.

That second sum is [itex]\sum_{i=1}^n \mu= n\mu[/itex] and that is the exponential
[tex] exp\left(\frac{n\lambda}{\mu}\right)[/tex]

Now, that denominator: is that
[tex]2^{\mu^2}[/tex]
or
[tex]2^{2\mu}[/tex]
or just [itex]2\mu^2[/itex]

It looks like it is supposed to be 2 to a power but to get that result it must be [itex]2\mu^2[/itex].
 

1. What is an exponential equation?

An exponential equation is a mathematical expression that involves a base number raised to a variable power. For example, y = 2^x is an exponential equation where 2 is the base and x is the variable power.

2. How is the exponential equation derived?

The exponential equation is derived by using the properties of logarithms and exponents. Specifically, it is derived from the definition of logarithms, which states that logba = c if and only if bc = a. This definition is used to solve for the variable in an exponential equation.

3. What is the significance of the base number in an exponential equation?

The base number in an exponential equation determines the rate of change of the equation. A larger base number will result in a steeper curve, while a smaller base number will result in a more gradual curve. The base number also determines if the equation will have a positive or negative growth or decay.

4. What are some real-life applications of exponential equations?

Exponential equations are used to model various natural phenomena, such as population growth, radioactive decay, and compound interest. They are also used in fields such as physics, biology, and economics to describe exponential growth or decay processes.

5. Can an exponential equation have a negative base?

No, an exponential equation cannot have a negative base. This is because a negative base results in a complex number when raised to a fractional power, which is not a valid solution for most real-world applications. However, exponential equations can have a negative exponent, which results in a fraction with a positive base.

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