Suppose I took measurements $\{X_i\}$, which are all independent and they follow a normal distribution $X_i\sim N(\mu,\sigma)$. I am asked for the joint probability of all of the measurements. Based on joint probability, I expect:
$$Prob(x_i=X_i)=\prod Prob(x_i)$$ $$=\prod\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
Where I use $\prod$ as a product over all $i$. Am I correct in saying this, or am I missing something important?
Yes you are correct, but small correction. $$ P(X_i= x_i) = \prod P(x_i)= \prod \frac{1}{\sigma \sqrt{2\pi}} e^{-(1/2\sigma^2)(x_i-\mu)^2} = \left(\frac{1}{\sigma \sqrt{2\pi}}\right)^n e^{-(1/2\sigma^2)\sum (x_i-\mu)^2} $$