Density funciton for minimum of $n$ independent variables I am not a mathematics guy, and the question which I am gonna ask would be pretty simple for mathematicians. In fact I am doing research and was reading some blogs. I wanted to derive the density function for $n$ number of independent variables. following relation I find on Internet for calculating CDF of $n$ number of independent variables
$$
F(T) = 1 – (1 - F_1(T)) (1 - F_2(T)) \dots (1- F_n(T)) = 1 – \prod_{i=1}^n F_i (T)
$$
 but for PDF they says that take the derivative of it . In fact I learned mathematics decades ago, so I don't know how to do that. Can some one just help me in steps how to take derivate of above equation and finally what would be the Density function?
 A: Take logarithm and then take derivatives.
A: Since the n variables are independent you have that $F_{X_1,X_2\cdots X_n}(x_1,x_2,\cdots,x_n)=\mathbb{P}\{X_1\leq x_1, X_2\leq x_2\cdots ,X_n\leq x_n\}=\mathbb{P}\{X_1\leq x_1\}\cdots \mathbb{P}\{X_n\leq x_n\}=F_{X_1}(x_1)\cdot F_{X_2}(x_2)\cdots F_{X_n}(x_n)$. 
Now, by the definition of density and cumulative distribution you have that
$F_{X_1,X_2\cdots X_n}(x_1,x_2,\cdots,x_n)=\int_{-\infty}^{x_1}\int_{-\infty}^{x_2}\cdots \int_{-\infty}^{x_n} f_{X_1,X_2\cdots X_n}(t_1,t_2,\cdots,t_n)dt_1dt_2\cdots dt_n$. On the other hand
$F_{X_1}(x_1)\cdot F_{X_2}(x_2)\cdots F_{X_n}(x_n)=\int_{-\infty}^{x_1}f_{X_1}(t_1)dt_1\cdot\int_{-\infty}^{x_2}f_{X_2}(t_2)dt_2\cdots\int_{-\infty}^{x_n}f_{X_n}(t_n)dt_n=\int_{-\infty}^{x_1}\int_{-\infty}^{x_2}\cdots \int_{-\infty}^{x_n} f_{X_1}(t_1)f_{X_2}(t_2)\cdots f_{X_n}(t_n)dt_1dt_2\cdots dt_n.$ Because the left hand sides we showed earlier are equal, it follows that the joint density function of independent RVs is the product of the individual density functions. Hence, no need to differentiate the cdf. Hope this helps. 
