Question

Let the random variable X be the number of rooms in a randomly chosen owner-occupied housing unit in a certain city. Here are the distributions of the number of rooms for owner-occupied units and renter-occupied units:

Rooms 1 2 3 4 5
Owned 0.003 0.002 0.023 0.105 0.218
Rented 0.008 0.027 0.287 0.362 0.165
Rooms 6 7 8 9 10
Owned 0.225 0.198 0.149 0.053 0.024
Rented 0.095 0.033 0.013 0.003 0.004

Required:
a. Express "the unit has 7 or more rooms" in terms of X. x≥7. What is the probability of this event?
b. Express the event {X > 7} in words.
c. What important fact about discrete random variables does comparing your answers to (a) and (b) illustrate?

Answer 1

Part a:

"The unit has 7 or more rooms" in terms of X is given by the unit which has 7, 8, 9 or 10 rooms.

P{ X≥7}= 0.424

Part b:

The event {X > 7} in words is that the unit has more than 7 rooms. .

Part C:

A continuous random variable is given by intervals of numbers

A discrete random variable is the variable that is obtained by counting.

Explanation:

Part a:

"The unit has 7 or more rooms" in terms of X is given by

{ X≥7} = {x=7} + {x=8} + {x=9} + {x=10}

the unit which has 7, 8, 9 or 10 rooms.

The probability of this event is given by

P{ X≥7} =P {x=7} + P {x=8} + P {x=9} + P {x=10}

P{ X≥7} = 0.198 + 0.149+ 0.053 + 0.024

P{ X≥7}= 0.424

Part b:

The event {X > 7} in words is that the unit has more than 7 rooms. This does not include the unit having 7 rooms.

Part C:

A continuous random variable takes all the values in the given interval as in part a.The random variable X takes the values between 7 and 10 inclusive of 7.

A discrete random variable is the variable that is obtained by counting as in part b. The variable X takes the values of the units with 8 , 9 or 10 rooms.