Question

Suppose that the length of long distance phone calls, measured in minutes, is known to have an exponential distribution with the average length of a call equal to eight minutes.

a. Define the random variable. $X=$

b. Is X continuous or discrete?

c. $X~$

d. $\mu =$

e. $\sigma =$

f. Draw a graph of the probability distribution. Label the axes.

g. Find the probability that a phone call lasts less than nine minutes.

h. Find the probability that a phone call lasts more than nine minutes.

i. Find the probability that a phone call lasts between seven and nine minutes.

j. If 25 phone calls are made one after another, on average, what would you expect the total to be? Why?

Short answer

(a) $\mathrm{X}=$The length of long distance phone calls.

(b) $X$is Continuous.

(c) $\mathrm{X}~Exp\left(\frac{1}{8}\right)$

(d) The required value is $\mu =8$.

(e)The standard deviation is $\sigma =8$

(f) The required graph is

(g) The chances of a phone call lasting less than nine minutes are slim $P(X<9)=0.6753$.

(h) The likelihood that a phone call will go longer than nine minutes is $P(X>9)=0.3247$.

(i) The probability that a phone call lasts between seven and nine minutes is $\mathrm{P}(7<\mathrm{x}<9)=0.0922$.

(j) if 25 calls are made so the total call duration will be 200 minutes.

Step-by-step solution

Part (a) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (a)  Step 2: Define the random variable X.

The random variable, $\mathrm{X}$can be defined as,

$\mathrm{X}=$The duration of long-distance calls.

Part (b) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (b) Step 2: X is continuous or discrete.

Because the random variable X has an exponentially distributed distribution, X will be continuous.

Part (c) Step 1: Given Information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (c) Step 2:  Find the value of X~

According to the information provided, the random variable X has an exponential distribution.

$$\begin{gathered} \mathrm{X}~Exp\left(\frac{1}{8}\right) \\ \mathrm{X}~Exp(0.125) \end{gathered}$$

Part (d) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (d) Step 2: Find the value of μ.

The formula for calculating the mean is as follows:

$\mu =8.$

Part (e) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (e) Step 2:  Find the Standard deviation.

The formula for calculating standard deviation is:

$$\begin{gathered} \sigma =\mu \\ \sigma =8 \end{gathered}$$

Therefore standard deviation is $\sigma =8$.

Part (f) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (f) Step 2:Create a probability distribution graph. Axes should be labelled.

We know that,

$X~Exp(0.125),\mathrm{m}=0.125$

Probability density function of exponential distribution is given by,

$$\begin{gathered} f\left(x\right)=m{e}^{-mx} \\ f\left(x\right)=0.125e^{-0.125x} \end{gathered}$$

From above equation which represents the probability density function, make the graph by using the above obtained probability density function.

The value of maximum f(x) will lie on y axis and at x=0 is given by,

$$\begin{gathered} \mathrm{f}\left(0\right)=m \\ \mathrm{f}\left(0\right)=0.125{\mathrm{e}}^{-0.125\times 0} \\ \mathrm{f}\left(0\right)=0.125{\mathrm{e}}^0 \\ \mathrm{f}\left(0\right)=0.125 \end{gathered}$$

Graph can be written as below,

Part (g) step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (g) step 2:  Find the probability that phone call lasts less than nine minutes.

Calculate the required probability using,

$$\begin{gathered} P(X<x)=1-m{e}^{-mx} \\ P(X<9)=1-e^{-0.125\times 9} \\ P(X<9)=1-0.3247 \\ P(X<9)=0.6753 \end{gathered}$$

Therefore , The chances of a phone call lasting less than nine minutes are slim. $P(X<9)=0.6753$.

Part (h) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (h) Step 2: Determine the likelihood that the phone call will last longer than nine minutes.

The required probability can be calculated using the following formula:

$$\begin{gathered} P(X>x)=1-P(x<9) \\ P(X>9)=1-0.6753 \\ P(X>9)=0.3247 \\ P(X<9)=0.6753 \end{gathered}$$

Therefore the probability that phone call lasts more than nine minutes $P(X>9)=0.3247$.

Part (i) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (i) Step 2:  Calculate the likelihood of a phone call lasting between seven and nine minutes.

The required probability can be calculated using the following formula:

$$\begin{gathered} \mathrm{P}(7<\mathrm{x}<9)=\mathrm{P}(\mathrm{x}<9)-\mathrm{P}(\mathrm{x}<7) \\ \mathrm{P}(7<\mathrm{x}<9)=1-{\mathrm{e}}^{-0.125\times 9}-\left(1-{\mathrm{e}}^{-0.125\times 7}\right) \\ \mathrm{P}(7<\mathrm{x}<9)={\mathrm{e}}^{-0..875}-{\mathrm{e}}^{-1.125} \\ \mathrm{P}(7<\mathrm{x}<9)=0.4169-0.3247 \\ \mathrm{P}(7<\mathrm{x}<9)=0.0922 \end{gathered}$$

As a result, the likelihood of a phone call lasting between seven and nine minutes is high. $\mathrm{P}(7<\mathrm{x}<9)=0.0922$

Part (j) Step 1: Given information

The average time of a long distance phone call is eight minutes, according to an exponential distribution of call lengths measured in minutes.

Part (j) Step 2:  What would you estimate the total to be if 25 phone calls were made one after the other, and why?

The average length of a phone call is 8 minutes, thus if 25 calls are made, the total call time will be.

$25\times 8$

$=200minutes$