Working with the Uniform distribution and the Exponential distribution (Business)

Number of Questions:	2
Marks Available:	6
Pass Percentage:	0%
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Question 1

The time, in minutes between customer arrivals at the RyanJet check-in desk at Newcastle Airport follows an exponential distribution with rate $\var{ra}$ i.e.

$X \sim \operatorname{Exp}(\var{ra})$

a)

Find $\operatorname{E}[X]$ between customer arrivals at the RyanJet check-in desk at Newcastle Airport :

$\operatorname{E}[X]=\;$ ? Expected answer: minutes (enter as a decimal correct to 3 decimal places).

Find $\operatorname{Var}(X)$ :

$\operatorname{Var}(X)=\;$ ? Expected answer: (enter as a decimal correct to 3 decimal places).

This feedback is on your last submitted answer. Submit your changed answer to get updated feedback.

Score: 0/2

Unanswered

b)

Find the probability that the time between two customers arriving is less than $\var{thistime}$ minutes:

$P(X \lt \var{thistime})=\;$ ? Expected answer: (enter as a decimal correct to 3 decimal places)

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Score: 0/1

Unanswered

Advice

If $X \sim \operatorname{Exp}(\lambda)$ then $\displaystyle \operatorname{E}[X] =\frac{1}{\lambda}$ and $\displaystyle \operatorname{Var}(X)=\frac{1}{\lambda^2}$ .

Also $P(X \lt a)=1-e^{-\lambda a}$ .

If $X \sim \operatorname{Exp}(\var{ra})$ then:

$\displaystyle \operatorname{E}[X] =\frac{1}{\lambda}=\frac{1}{\var{ra}}=\var{ans1}$ to 3 decimal places.

$\displaystyle \operatorname{Var}(X) =\frac{1}{\lambda^2}=\frac{1}{\var{ra}^2}=\var{ans2}$ to 3 decimal places.

$P(X \lt \var{thistime}) = 1 -(e ^ {-\var{ ra} \times \var{thistime}}) = 1 -(e ^ { -\var{ra * thistime}}) = \var{ans3}$ to 3 decimal places.

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Question 2

A new supermarket plans to open somewhere on the outskirts of a town. In fact, $X$, the distance of a new supermarket from the town centre is Uniformly distributed between $\var{lower}$ metres and $\var{upper}$ metres i.e.

\[X \sim \operatorname{U}(\var{lower},\var{upper})\]

a)

Find $\operatorname{E}[X]$, the expected distance in metres of the new supermarket from the town centre:

$\operatorname{E}[X]=\;?$ Expected answer: m (to 3 decimal places).

Also find the variance $\operatorname{Var}(X)$:

$\operatorname{Var}(X)=\;$? Expected answer: (to 3 decimal places).

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Score: 0/2

Unanswered

b)

Find the probability that the supermarket opens within $\var{thisdis}$ kilometres of the town centre.

$P(X \le \var{thisdis}\textrm{km})=\;$? Expected answer: (to 3 decimal places).

This feedback is on your last submitted answer. Submit your changed answer to get updated feedback.

Score: 0/1

Unanswered

Advice

a) For a Uniform distribution \[X \sim \operatorname{U}(\var{lower},\var{upper})\] we have:

$\displaystyle \operatorname{E}[X] = \frac{\var{lower}+\var{upper}}{2}=\var{ans1}$m

$\displaystyle \operatorname{Var}[X] = \frac{(\var{upper}-\var{lower})^2}{12}=\frac{(\var{upper-lower})^2}{12}=\var{ans2}$ to 3 decimal places.

$\displaystyle P(X \le \var{thisdis}\textrm{km})=\frac{\var{thisdis}\times 1000 -\var{lower}}{\var{upper}-\var{lower}}=\var{ans3}$ to 3 decimal places.

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Working with the Uniform distribution and the Exponential distribution (Business)

Question 1

a)

b)

Advice

Question 2

a)

b)

Advice