Joint Distribution of iid Random Variables

Sep 2011
393
8
Mumbai (Bombay),Maharashtra State,India
Hi,
Let X,Y,Z be independent, identically distributed random variables,each with density$f(x)=6x^5$ for $0 \leq x\leq 1$ and 0 elsewhere.I want to find the distribution and density functions of the maximum of X,Y and Z.
Answer:-I know by integrating density function, distribution function can be obtained.But I am not understanding how to find maximum of X,Y and Z. Will any member guide me in this regard?
 
Last edited:
Jul 2011
1,254
389
Belgium
Hi,
Let X,Y,Z be independent, identically distributed random variables,each with density$f(x)=6x^5$ for $0 \leq x\leq 1$ and 0 elsewhere.I want to find the distribution and density functions of the maximum of X,Y and Z.
Answer:-I know by integrating density function, distribution function can be obtained.But I am not understanding how to find maximum of X,Y and Z. Will any member guide me in this regard?
Denote $M = \max\{X,Y,Z\}$ the maximum of the three iid random variables $X,Y$ and $Z$, and let $F_M$ be the distribution function of $M$. By definition:
$$F_M(x) = \mathbb{P}(M \leq m).$$
To proceed, think about the following: the maximum $M$ is smaller than $m$ if and only if $X$, $Y$ and $Z$ are smaller than $m$. Therefore :
$$\mathbb{P}(M \leq m) = \mathbb{P}(X \leq m, Y \leq m, Z \leq m) = \mathbb{P}(X \leq m)\mathbb{P}(Y \leq m)\mathbb{P}(Z \leq m),$$
where the last equality is a consequence of the independency of $X$, $Y$ and $Z$.
 
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