Mathematics: The random variable X is exponentially distributed, where X represents the waiting time to be seated at a restaurant during the evening. If X has an average value of 19 minutes, what is the probability that the wait time is greater than 14 minutes

The random variable X is exponentially distributed,

Ответ:

HaseelM

20.01.2022

Mathematics

The probability that the wait time is greater than 14 minutes is 0.4786.

Step-by-step explanation:

The random variable X is defined as the waiting time to be seated at a restaurant during the evening.

The average waiting time is, β = 19 minutes.

The random variable X follows an Exponential distribution with parameter $\lambda=\frac{1}{\beta}=\frac{1}{19}$ .

The probability distribution function of X is:

$f(x)=\lambda e^{-\lambda x};\ x=0,1,2,3...$

Compute the value of the event (X > 14) as follows:

$P(X14)=\int\limits^{\infty}_{14} {\lambda e^{-\lambda x}} \, dx=\lambda \int\limits^{\infty}_{14} {e^{-\lambda x}} \, dx\\=\lambda |\frac{e^{-\lambda x}}{-\lambda}|^{\infty}_{14}=e^{-\frac{1}{19} \times14}-0\\=0.4786$

Thus, the probability that the wait time is greater than 14 minutes is 0.4786.

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The random variable X is exponentially distributed, where X represents the waiting time to be seated at a restaurant during the evening. If X has an average value of 19 minutes, what is the probability that the wait time is greater than 14 minutes

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