According to the Census Bureau publication Current Population Reports, the probability distribution for household size (number of people per household, say X) in the United States is as follows. For the purpose of the next two questions, the experiment consists of randomly selecting a household in the United States and observing the number of people living there.
x 1 2 3 4 5 6 7
P(X = x) 0.16 0.32 0.05 0.31 0.10 0.05 0.01
How many people can we expect to see living in the randomly chosen household?
Answer
4.14
3.06
4.00
1.79
1.24
What is the probability that the randomly selected household will contain more than 4 people?
Answer
0.47
0.31
0.16
0.84
0.53
To find the expected number of people living in the randomly chosen household, you need to calculate the weighted average of the number of people (x) multiplied by their corresponding probabilities (P(X=x)).
Expected Value (μ) = Σ(x * P(X=x))
Using the given probability distribution:
Expected Value = (1 * 0.16) + (2 * 0.32) + (3 * 0.05) + (4 * 0.31) + (5 * 0.10) + (6 * 0.05) + (7 * 0.01)
Calculating this expression will give you the answer.
For the probability that the randomly selected household will contain more than 4 people, you need to sum up the probabilities of all the households with more than 4 people.
P(X > 4) = P(X = 5) + P(X = 6) + P(X = 7)
Again, using the given probability distribution, substitute the values and calculate the expression to find the answer.
To find the correct answers, you should perform these calculations.