- #1

mathmari

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The burning time of an electric candle is between $60$ and $80$ hours. It is considered to be continuously uniformly distributed over this period. A Christmas tree with $12$ candles (independent of the burning time), only lights up as long as all candles are working.

Determine, with intermediate steps, for the "burn time" of the tree, the distribution function and the average burn duration.

Hint : To do this, model the burning time of the individual candles using suitable independent and identically distributed random variables $X_1, X_2,\ldots , X_{12}$. The burn duration of the tree is then given by $Y = \min \{X_1, X_2,\ldots , X_{12}\}$. First determine $P [Y \geq x]$ and note that the minimum of $12$ real numbers is greater than or equal to $x$ when all $12$ numbers are greater than or equal to $x$.

Could you explain to me why $Y$ is defined as the minimum of all $X_i$'s ?

We have to calculate the distribution function $F_Y(y)$ and the average burn duration, i.e. the expected value $E[Y]$, right?

:unsure: