https://file.aiquickdraw.com/mathbot/file/172856142303147ukjo1i.jpg

The problem asks you to:

(i) **Construct the probability distribution** for the number of cars \(X\) in a household, based on the given information.  
(ii) **Find the expected value** of \(X\).

### (i) Construct the probability distribution
We are given the following information about the distribution of cars per household:
- 25% of households have 1 car.
- 45% of households have 2 cars.
- 10% of households have 3 cars.
- The rest of the households have no cars.

The probability distribution \( P(X = x) \) is summarized as follows:

\[
\begin{array}{c|c}
X & P(X) \\
\hline
0 & 1 - (0.25 + 0.45 + 0.10) = 0.20 \\
1 & 0.25 \\
2 & 0.45 \\
3 & 0.10 \\
\end{array}
\]

Thus, the probability distribution is:
- \( P(X = 0) = 0.20 \)
- \( P(X = 1) = 0.25 \)
- \( P(X = 2) = 0.45 \)
- \( P(X = 3) = 0.10 \)

### (ii) Find the expected value of \(X\)

The expected value \(E(X)\) is calculated using the formula for the expectation of a discrete random variable:

\[
E(X) = \sum_{x} x \cdot P(X = x)
\]

Substituting the values:

\[
E(X) = (0 \cdot 0.20) + (1 \cdot 0.25) + (2 \cdot 0.45) + (3 \cdot 0.10)
\]

\[
E(X) = 0 + 0.25 + 0.90 + 0.30 = 1.45
\]

Thus, the expected value of \(X\) is **1.45 cars per household**.

Would you like more details or have any other questions?

Here are five related questions you can explore:
1. How is the variance of a probability distribution calculated?
2. What does the expected value represent in real-world scenarios?
3. How do you construct a cumulative distribution from the given data?
4. How would the expected value change if more households had 3 cars?
5. What if we introduced a household with 4 cars? How would that affect the distribution?

**Tip**: The expected value is not always a number that appears in the data but rather the "average" outcome over many trials.

In a residential area, 25% of the household is having one car, 45% of the household is having two cars, 10% of the household is having three cars and the rest of the household has no cars. By using X as the number of cars in a household, (i) construct the probability distribution for the situation above, and (ii) find the expected value of X.

Learn how to construct a probability distribution and find the expected value of the number of cars in a household. This problem involves key probability concepts and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation