Can we solve this equation with pdf and cdf?

**MilosM** · 12-27-2011 01:59 PM

Dear,
suppose we have,
pdf((x-1)/2)+cdf((x-3)/4)=3
where pdf is a normal probability density function
and cdf is cumulative density function.
How to express x from this equation?
Best,
Milos

**ledzep** · 12-27-2011 02:15 PM

I can't see why you can't compute this (theoretically). Just substitute the values you want to compute at the pdf and cdf. To express x, you just have to rearrange and make x the subject.

Typing up Normal pdf and Normal CDF will take a bit of time. So to get home the message , let me make a fictitious cdf and pdf.

Suppose,

$F(x)=x^2$ is your CDF

$\Rightarrow F(\frac{x-3}{4})=(\frac{x-3}{4})^2$

Using your CDF, you work out your pdf. You differentiate CDF to get your pdf.

In this example,
$f(x)= 2*x$ is your pdf

$\Rightarrow f(\frac{x-1}{2})=2*(\frac{x-1}{2})$

So, the task is just to solve:

$2*(\frac{x-1}{2}) + (\frac{x-3}{4})^2 = 3$

So, this is just some basic algebra.

Sometimes, you may not be able to solve the equation explicitly, in which case you can resort to numerical methods.

HTH.

**Dragan** · 12-27-2011 08:43 PM

Originally Posted by MilosM

Dear,
suppose we have,
pdf((x-1)/2)+cdf((x-3)/4)=3
where pdf is a normal probability density function
and cdf is cumulative density function.
How to express x from this equation?
Best,
Milos

Your equation won't work, in general, for any normal distribution. For example, simply consider the standard normal distribution.

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