For r.v. Z = max(X1, ..., Xn), what is $f_Z(z)$, given X1, X2, ..., Xn are independent.

Question

I'm trying to understand the problem in the picture above.

What happened to the "dz" between this step:

$$f(Z)~ dz~ \Bigg( \int \limits_{-\infty}^{z} f(x)~dx \Bigg)^{n-1}$$

and this step:

$$f_Z(z) = 2~f(z) \int \limits_{-\infty}^{z} f(x)~dx = 2~f(z)~F(z)$$

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A few properties that I know:

• for continuous r.v. $P(X=x) = 0$
• $P(a < X \le b) = \int \limits_{b}^{a} f_X(x)~dx$
• Max(a, b) = (a > b) ? a : b
• Max(a, b, c)

  Max(a,b,c) =   
    if (a >= b) and (a >= c): 
        largest = a 

    elif (b >= a) and (b >= c): 
        largest = b 
    else: 
        largest = c 

    return largest 

Answer 1

RULE: Converting infinitesimal probability into PDF

When converting an infinitesimal probability into a PDF, remove the dx:

$$\underbrace{P(x < X < x + dx) = f_X(x)~dx}_\text{infinitesimal probability} \longleftrightarrow \underbrace{f_X(x)}_\text{PDF}$$