Respuesta :
Answer:
Probability that the wait time is greater than 37 minutes is 0.3474.
Step-by-step explanation:
We are given that the random variable X is known to be exponentially distributed and X be the waiting time for a car to pass by on a country road, where X has an average value of 35 minutes.
Let X = waiting time for a car to pass by on a country road
The probability distribution function of exponential distribution is given by;
[tex]f(x) = \lambda e^{-\lambda x} , x >0[/tex] where, [tex]\lambda[/tex] = parameter of distribution.
Now, the mean of exponential distribution is = [tex]\frac{1}{\lambda}[/tex] which is given to us as 35 minutes that means [tex]\lambda = \frac{1}{35}[/tex] .
So, X ~ Exp( [tex]\lambda = \frac{1}{35}[/tex] )
Also, we know that Cumulative distribution function (CDF) of Exponential distribution is given as;
[tex]F(x) = P(X \leq x) = 1 - e^{-\lambda x}[/tex] , x > 0
Now, Probability that the wait time is greater than 37 minutes is given by = P(X > 37 min) = 1 - P(X [tex]\leq[/tex] 37 min)
P(X [tex]\leq[/tex] 37 min) = [tex]1 - e^{-\frac{1}{35} \times 37}[/tex] {Using CDF}
= 1 - 0.3474 = 0.6525
So, P(X > 37 min) = 1 - 0.6525 = 0.3474
Therefore, probability that the wait time is greater than 37 minutes is 0.3474.
