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Let us say we have $X$ and $Y$ two independent random variables.
How can we show $Z = min(X, Y)$ or $Z=max(X, Y)$ mathematically to find the mean and pdf of $Z$?

Atif
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1 Answers1

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This is the proceeding for $Z=\max(X,Y)$

$$F_Z(z)=\mathbb{P}[Z\le z]=\mathbb{P}[X\le z,Y\le z]=\dots\text{ using independence }\dots=\mathbb{P}[X\le z]\cdot\mathbb{P}[Y\le z]=F_X(z)\cdot F_Y(z)$$

can you do a similar reasoning for $U=\min(X,Y)$?

tommik
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  • I did not understand how you can simply do and in place of max? Also, I only have X and Y. how can I evaluate $F_X(z)$ and $F_Y(z)$ without knowing $Z$? The question is how can I write $Z$ using $X$ and $Y$? – Atif Nov 02 '21 at 19:01
  • Please read well what I wrote. Here $z$ is simply a value, not a rv – tommik Nov 02 '21 at 19:22
  • Here it is for $U = min(X,Y)$, $$F_Z(z)=\mathbb{P}[Z\le z]=\mathbb{P}[X\ge z,Y\ge z]=\dots\text{ using independence }\dots=\mathbb{P}[X\ge z]\cdot\mathbb{P}[Y\ge z]=(1-F_X(z))\cdot (1-F_Y(z))$$ – Atif Nov 02 '21 at 19:41
  • You are almost there. You found $P(Z>z)$.Thus $F_Z(z)$ is the complement. $F_Z(z)=1-(1-F_X(z))(1,F_Y(z))$ – tommik Nov 02 '21 at 19:44
  • how dies min and max relate to $\le, \ge$ signs? – Atif Nov 02 '21 at 19:46
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    Observe that if max is < z then all rv are themselves <z. In the same way, if min is greater than u, all rv are themselves >u – tommik Nov 02 '21 at 19:51
  • Let us continue this discussion in chat. – Atif Nov 02 '21 at 20:00