A residential family is considering two cities for relocation. The data related to pollutant exposure and associated health cost per year are given in the following figure.
The pollutant exposure is characterized in high, mild and low exposure categories with respective probability values. The difference in expected value of health cost of City1 with respect to that of City 2 is ________ lakhs/year. (rounded off to two decimal places).
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The expected value of a discrete random variable is calculated as the sum of the product of each possible value and its probability:
\[
E(X) = \sum x_i P(x_i)
\]
Step 1: Calculate the expected health cost for City 1.
The expected health cost for City 1 is calculated by multiplying each possible health cost by its corresponding probability and summing these values:
\[
E({Cost}_{{City 1}}) = (0.1 \times 10) + (0.5 \times 4) + (0.4 \times 8)
\]
\[
E({Cost}_{{City 1}}) = 1 + 2 + 3.2 = 6.2 { lakhs/year}
\]
Step 2: Calculate the expected health cost for City 2.
The expected health cost for City 2 is calculated similarly:
\[
E({Cost}_{{City 2}}) = (0.2 \times 8) + (0.8 \times 1.5)
\]
\[
E({Cost}_{{City 2}}) = 1.6 + 1.2 = 2.8 { lakhs/year}
\]
Step 3: Calculate the difference in the expected health cost of City 1 with respect to that of City 2.
The difference is given by:
\[
{Difference} = E({Cost}_{{City 1}}) - E({Cost}_{{City 2}})
\]
\[
{Difference} = 6.2 - 2.8 = 3.4 { lakhs/year}
\]
Rounding off to two decimal places, the difference is 3.40 lakhs/year.
