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Using simulation, I have estimated the probability density function ($F$) of a random variable ($X$). Now I measure the value of one instance ($x$) and the standard deviation of it. The uncertainty of the measurement is approximated with a normal distribution ($N$). If I'm not wrong, the probability that $x$ is between $a$ to $b$ is $\int_a^bNF \;dx$. This should be a very common procedure. But I cant remember anywhere I have seen something like this example, thus don't know what keywords I should search for.

Excessive googling with anything related I could think of didn't help. It seems a silly question, but do know where I can find examples similar to this? or how this process is described or some pointers?

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    @DilipSarwate, thanks. I would really appreciate it if you could give a hint why that integral expression is wrong. I didn't explain these are physical measurements and as I mentioned I do estimate the uncertainty of each measurement. That's what makes this different from simple examples in the probability books I have read. Example 6b you mentioned is really simple, and I cant find the connection between these two problems. Thanks again.2012-04-28

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$x$ will be sampled from a distribution whose probability density function is $F$. $\:$ $y$ will be
sampled from a normal distribution with mean $x$ and standard deviation $\sigma$. $\:$ Let $N$ be the probability density function for a normal distribution with mean $y$ and standard deviation $\sigma$.

Given $y$, the probability that $x$ is between $\:a\:$ and $\:b\:$ is

$\frac1{\displaystyle\int_{-\infty}^{\scriptsize+\normalsize\infty} (N(x) \cdot F(x))\;dx} \cdot \left(\displaystyle\int_a^b (N(x) \cdot F(x))\;dx\right) \;\;\;$.



http://en.wikipedia.org/wiki/Bayes%27_theorem

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    The product is the likelihood of the observation given that value, times the initial likelihood of that value. $\:$ The product is the likelihood (i.e., a constant times the probability density) function of $x$. $\hspace{.6 in}$2012-04-28