Combination of probabilities for probability densities

Question

If I have an experiment with discrete probabilities like drawing black and white balls, the probability of drawing both black and white is just a multiplication of probabilities of drawing black and white. Similarly, if I want to work with "OR" type of situations, I add the probabilities.

But how does this work with continuous random variables? Say, I have the following experiment: I have a large bucket of some material and I am trying to determine the melt temperature. So I take small samples and put them on a burner and note the temperature when they melt. Let's assume that after many of such measurements, I get a gaussian with some mean value and variance.

If I then would like to say something about another bucket of the same material, I might ask, what is the probability of a sample melting at temperature between $(T,T+{\rm d}T)$. Then the probability should be $\Pr(\text{melt at }T)\approx f(T)\,{\rm d} T$ or more exactly $\int_T^{T+{\rm d}T}f(x)\,{\rm d} x$ where $f(T)$ is the gaussian.

But what if I want to ask about two samples from taken from the bucket both melting at $\{T,T+{\rm d}T\}$ or one of the samples from the bucket melting between $(T_1,T_1+{\rm d}T)$ and the other from the bucket melting between $(T_2,T_2+{\rm d}T)$? Is it just $f^2(T)\,{\rm d}x^2$ and $f(T_1)f(T_2)\,{\rm d}x^2$ respectively? Or more exactly

$$\int_T^{T+{\rm d}T}f(x)\,{\rm d} x\int_T^{T+{\rm d}T}f(x)\,{\rm d} x$$

or

$$ \int_{T_1}^{T_1+{\rm d}T} f(x){\rm d} x \int_{T_2}^{T_2+{\rm d}T}f(x)\,{\rm d} x $$

But this leads to having new probability distribution functions $f(x)f(y)$ which from what I have read are not necessarily distribution functions.

And if I want to ask about the probability of the melt temperature being between $(T_1, T_1+{\rm d}T)$ or between $(T_2,T_2+{\rm d}T)$, would it be $f(T_1)\,{\rm d}T + f(T_2)\,{\rm d}T$ or more precisely

$$\int_{T_1}^{T_1+{\rm d}T} f(x)\,{\rm d} x + \int_{T_2}^{T_2+{\rm d}T} f(x)\,{\rm d} x\text{?}$$

Answer 1

I think that there is an issue about terminology (which is my feeble defense about my earlier interpretation of the problem).

If the temperature of a single sample can be considered to have a Gaussian distribution with mean $\mu$ and variance $\sigma^2$, then the probability of the observed temperature being between $T$ and $T+dT$ is given by

$$p = \Phi\left({{T+dT-\mu}\over{\sigma}}\right) - \Phi\left({{T-\mu}\over{\sigma}}\right)$$

The probability of two independent samples both being between $T$ and $T+dT$ is the same temperature range is the square of the above ($p^2$). If the samples are not independent, then you'd need to define how they are dependent. Also, this is not the same answer to the question "What is the probability that both samples will be within $dT$ of each other?"

Still assuming independence of samples and that you know what the parameters are for the Gaussian distribution, the probability that sample 1 is between $T_1$ and $T_1+dT$ and that sample 2 is between $T_2$ and $T_2+dT$ is just the product the two probabilities.

If now you want to know the probability of a single sample resulting in a temperature between $T_1$ and $T_1+dT$ or $T_2$ and $T_2+dT$, then the probability of that event occurring is just the sum of the two probabilities if there is no overlap in the two intervals.

Now suppose there is overlap with $T_1 \le T_2 \le T_1+dT$. Then the probability is

$$\Phi\left({{T_2+dT-\mu}\over{\sigma}}\right) - \Phi\left({{T_1-\mu}\over{\sigma}}\right)$$

But all of the above assumes that you know $\mu$ and $\sigma^2$. If you just have estimates of $\mu$ and $\sigma^2$ from previous samples, then you can certainly estimate the above probabilities but you'd also want to estimate some associated measure of precision. That would require a bit more explanation.