Question

The manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 8 minutes. What is the probability that a car can complete the transaction in less than 5 minutes

Answer 1

Answer:

Probability that a car can complete the transaction in less than 5 minutes is 0.465.

Step-by-step explanation:

We are given that the manager of a gas station has observed that the times required by drivers to fill their car's tank and pay are quite variable. In fact, the times are exponentially distributed with a mean of 8 minutes.

Let X = times required by drivers to fill their car's tank and pay

The probability distribution function of exponential distribution is given by;

$f(x) = \lambda e^{-\lambda x} , x >0$     where, $\lambda$ = parameter of distribution.

Now, the mean of exponential distribution is = $\frac{1}{\lambda}$  which is given to us as 8 minutes that means  $\lambda = \frac{1}{8}$  .

So, X ~ Exp( $\lambda = \frac{1}{8}$ )

Also, we know that Cumulative distribution function (CDF) of Exponential distribution is given as;

$F(x) = P(X \leq x) = 1 - e^{-\lambda x}$  , x > 0

Now, Probability that a car can complete the transaction in less than 5 minutes is given by = P(X < 5 min)

  P(X < 5 min) = $1 - e^{-\frac{1}{8} \times 5}$        {Using CDF}

                       = 1 - 0.5353 = 0.465

Therefore, probability that a car can complete the transaction in less than 5 minutes is 0.465.

Answer 2

Answer:

The probability that a car driver can complete the transaction in less than 5 minutes is 0.4647.

Step-by-step explanation:

Let X = times required by drivers to fill their car's tank and pay.

The average time required by drivers is, β = 8 minutes.

The random variable X is Exponentially distributed with parameter, $\lambda=\frac{1}{\beta}=\frac{1}{8}=0.125$.

The probability distribution function of exponential distribution is:

$f_{X}(x)=\lambda e^{-\lambda x};\ x>0$

Compute the probability that a car driver can complete the transaction in less than 5 minutes as follows:

$P(X$

Thus, the probability that a car driver can complete the transaction in less than 5 minutes is 0.4647.