Question

# Let X_{1},X_{2},...,X_{6} be an i.i.d. random sample where each X_{i} is a continuous random variable with probability density function f(x)=e^{-(x-0)}, x>0 Find the probability density function for X_{6}.

Random variables
Let $$X_{1},X_{2},...,X_{6}$$ be an i.i.d. random sample where each $$X_{i}$$ is a continuous random variable with probability density function
$$f(x)=e^{-(x-0)}, x>0$$
Find the probability density function for $$X_{6}$$.

2021-05-13
Step 1
We know,
$$f(x)=e^{-(x-0)},x>0$$
And,
$$X_{1}X_{2},...,X_{6}$$
are IID random variables
Step 2
Therefore,
$$f(X_{6})=e^{-(x_{6}-0)},x_{6}>0$$