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Let $X_1 , X_2, X_3, X_4, X_5$ be mutually independent lifetimes. Assume that each lifetime has the probability density function $$ f_X(x) = 2x,\quad 0<x<1.$$ The system is arranged as a five-component parallel system. Find the pdf of the system lifetime $Y=\max(X_1, X_2,...X_5)$.

I suppose the difficulty I'm having with this problem is the notion of a "five-component parallel system," as I have no idea what that is. Having tried a cursory google search of this topic and its relation to probability, I came up empty. My fourth grade study of circuits makes me think that this problem greatly involves conditional probability, yet without knowing what this question is asking I remain unsure.

Bad at algebra and proofs
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  • Hint: What is the probability all five lives are less than or equal to $y$? Then differentiate – Henry Jun 26 '17 at 23:55
  • @Henry4321 Would it be as simple as 1/10? But then I'm confused over the bounds, would they be as simple as negative infinity to infinity? – Bad at algebra and proofs Jun 27 '17 at 00:37
  • Try to adapt the method here: https://math.stackexchange.com/q/313390/9464 –  Jun 27 '17 at 01:05
  • @Jack While I had looked at that post before and had found it helpful in finding some example of how to use the product of the probabilities, I'm still skeptical of how to integrate the given pdf into the rest of the problem, that is, how to use the given $f_x(x)$. WHile that example was good for a general result, this complication has me wondering. – Bad at algebra and proofs Jun 27 '17 at 01:22

1 Answers1

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Note that $X_1,\cdots,X_5$ are i.i.d.

Step 1 $$ F_Y(y)=P(Y\leq y)=P(X_1\leq y,\cdots,X_5\leq y)=[P(X_1\leq y)]^5. $$

Step 2 Find $P(X_1\leq y)$ using the density of $X_1$ $$ P(X_1\leq y)=\int_{-\infty}^y2x\cdot 1_{[0,1]}(x)\, dx. $$ In this calculation, consider three cases separately: $y< 0$, $0\leq y\leq 1$, $y>1$. Explicitly you have when $0\leq y\leq 1$ $$ P(X_1\leq y)=2\int_0^yx\, dx=y^2, $$ which implies that $$ P(X_1\leq y)=\begin{cases} 0,&y<0\\ y^2,&0\leq y\leq 1\\ 1,&y\geq 1. \end{cases} $$

Step 3 Take the derivative of $F_Y$ to find the density.