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In order to switch on an engine a number of trials are made. Each trial lasts for $\tau$ seconds. The probability of trial being successful is $p$. find the distribution of the total time required to switch on the engine.

I figured that the number of trials required follow a geometric distribution which gives me the expected number of trials. But i have no clue how to write the distribution for the time taken.

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    $\begingroup$ The time will be a random variable that takes values $\tau, 2 \tau, 3 \tau. \ldots$ with probability $p, (1-p)p,(1-p)^2p, \ldots$. $\endgroup$
    – kmitov
    Commented Nov 14, 2017 at 11:14

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Let $T$ denotes the time taken to start the engine.
Let $X$ denotes the number of trials up to and including the trial that the engine is started.

We know that $T=\tau X$.

So now we just need to find $P(T=t)$. \begin{align} P(T=\tau t)&=P\left(\tau X=\tau t\right)\\ &=P(X=t)\\ &=(1-p)^{t-1}p\\ \end{align}

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