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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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A measurement or sample value or realization of a random variable is just a number; it has no standard deviation because it does not deviate from anything. In any case, your integral expression for the probability is wrong. You might want to Google for statistical estimation, or signal detection theory, or read a probability textbook such S. Ross, A First Course in Probability, Example 6b in Chapter 7.6 – Dilip Sarwate Apr 28 '12 at 2:33
@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. – Heslil Apr 28 '12 at 2:52
up vote 0 down vote accepted

$x$ 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

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

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Why are two density functions being multiplied, and what interpretation are we to assign to the product? – Dilip Sarwate Apr 28 '12 at 15:14
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 $x$. $\hspace{.6 in}$ – Ricky Demer Apr 28 '12 at 19:24

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