Question

In a certain city, the daily consumption of water (in millions of litres) is a random variable with probability density function given by:

𝟏 𝒆−𝒙 𝒇(𝒙)={𝟗𝟗 , 𝒙≥𝟎

𝟎

(i) Identify the distribution and use its parameter to find the city’s expected water consumption for any given day. [3 marks]

(ii) What is the probability that the water supply on a given day is inadequate if the daily capacity of the city is 9 million litres?

Answer 1

Let $m= \frac{1}{9}$

$f(x) = me^{-xm}, \;x≥0$

(i) This distribution is exponential with mean $= \frac{1}{m}=9$

Expected water consumption = 9 million liters.

(ii) CDF of exponential distribution $= P(X≤x)= 1 -e^{-mx}$

For inadequate supply, daily capacity should be less than 9 million liters.

$P(X≤9) = 1 -e^{- \frac{1}{9} \times 9} = 0.632$