Probability, expectation and variance of uniform distribution

Question 1

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).

Score: 0/2

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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).

Score: 0/1

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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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Probability, expectation and variance of uniform distribution

a)

b)

Advice