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Discrete Random Variables

How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.

Indicator Variables

The fine when caught speeding by a speed camera is \(90\) dollars, and that when caught running a red light is \(160\) dollars. If caught doing both at the same time, then the fine is twice the sum of the penalties of doing each. If \(D_{2i}\) and \(D_{3i}\) denote the indicator variables of speeding and running a red light, respectively, which of the following correctly represents the fine \(y_i\) (in dollars) the driver has to pay?

Suppose that the average initial monthly salary (in \(10^4\) KRW) of a company employee in South Korea is given by the formula: \[y_i=104+94D_{2i}+162D_{3i}+210D_{4i},\] where the default is educated up to high school, and \(D_{2i}, D_{3i},\) and \(D_{4i}\) are the indicator variables for having a bachelor's degree, master's degree, and PhD, respectively. What is the expected salary difference (in \(10^4\) KRW) between Roger who has a PhD and Cliff who has a bachelor's degree?

Suppose there is a parking lot where the parking fee is \(6\) dollars for the first hour, and costs an additional fee of \(5\) dollars for every additional hour. There is also an extra charge of \(10\) dollars for trucks, while compact cars get a 50% discount. If the random variable \(X\) represents the number of hours parked, and \(D_{2i}\) and \(D_{3i}\) denote the indicator variables for trucks and compact cars, respectively, which of the following represents the equation for the parking fee \(Y_i\) (in dollars)?

Note: \(X\) is calculated as the smallest integer that is larger or equal to the actual number of hours parked. For instance, if a car is parked for 2 and half hours, then \(X=3.\)

Suppose that the remaining life expectancy (in months) when diagnosed with lung cancer is given by the formula: \[y_i=54-11D_{2i}-29D_{3i}-47D_{4i},\] where \(D_{2i}, D_{3i},\) and \(D_{4i}\) are the indicator variables for stages 2, 3, and 4, respectively. If Ronald is diagnosed with lung cancer stage 3, how many more months is he expected to live?

Note: There are four stages in lung cancer, where increasing stage results in worse prognosis.

The entrance fee (in dollars) of a museum is given by the formula: \[y_i=15-7D_{2i}-8D_{3i},\] where \(D_{2i}\) and \(D_{3i}\) are the indicator variables for children (under 15 years of age) and the handicapped, respectively. What is the entrance fee for a 12-year-old handicapped child?

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