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A car wash facility washes cars in four steps-soap, rinse, dry and vacuum performed by one worker. The duration of each step is exponentially distributed with a mean of 9 minutes.

A car has to finish with all the four steps to enable the next car to begin the process. Find the probability that the car wash will take 30 minutes or less. Also, compute the expected length of the wash and the modal value.

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Let X is the random variable denoting dule'tn of car wash.

car wash in 4 stages.

$\therefore k = 4$

each stage is exponentially dist. with mean time = 9 minutes.

i.e. $KQ = \frac{1}{9}$

$\therefore$ The probability that the car wash will take 30 mins/less is given by

p ( x ≤ 30) = f (30)

$= 1 - \sum^k-1_i=0 \frac{e^-kqx (kqx)^i}{i !} 1 x \gt 0$

$= 1 - /sum^3_i\lt0 \frac{e^\frac{-1}{09^30} [ \frac{1}{09} \times 30]^i} {i !}$

= 1 - [ 0.0238 + 0.1192 + 0.1985 + 0.2203]

= 1 - 0.5738

= 0.4262

$E(x) = \frac{1}{0} = \frac{1}{1/60} = 60 mins$

$u(x) = \frac{1}{kq^2} = \frac{1}{4 (1/60)^2} = 900 mins^2$

$= \frac{k-1}{kq}$
$= \frac{4.1}{1/9}$