The question asks to find the expected value $E(X)$ for the random variable $X$, given its probability distribution as shown in the bar graph.

To calculate $E(X)$, we use the formula:

$E(X) = \sum_{x} x \cdot P(x),$ where:

• $x$ is a value of the random variable,
• $P(x)$ is the probability of $x$.

Steps to solve:

1. Identify all possible values of $x$ from the bar graph and their respective probabilities $P(x)$.
2. Multiply each $x$ by its corresponding $P(x)$.
3. Add up all the results to get $E(X)$.

Let me carefully examine the graph and calculate $E(X)$. One moment while I process the data in the image.From the provided image, we can interpret the bar graph as showing the probability distribution. Based on the heights of the bars, the values and probabilities are as follows:

Observed data:

• $x = 6$, $P(6) = 0.1$
• $x = 12$, $P(12) = 0.2$
• $x = 18$, $P(18) = 0.4$
• $x = 24$, $P(24) = 0.2$
• $x = 30$, $P(30) = 0.1$

Formula:

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

Calculation:

$E(X) = (6 \cdot 0.1) + (12 \cdot 0.2) + (18 \cdot 0.4) + (24 \cdot 0.2) + (30 \cdot 0.1)$

Performing the calculations step-by-step:

1. $6 \cdot 0.1 = 0.6$
2. $12 \cdot 0.2 = 2.4$
3. $18 \cdot 0.4 = 7.2$
4. $24 \cdot 0.2 = 4.8$
5. $30 \cdot 0.1 = 3.0$

Adding these: $E(X) = 0.6 + 2.4 + 7.2 + 4.8 + 3.0 = 18.0$

Final Answer:

$E(X) = 18.0$

Let me know if you’d like a detailed breakdown of the steps!

5 Related Questions:

1. How do you interpret the expected value in a real-world scenario?
2. What happens to $E(X)$ if all probabilities $P(x)$ are equal?
3. How do we calculate the variance and standard deviation of a probability distribution?
4. How would $E(X)$ change if $P(18)$ was increased to 0.5?
5. What is the importance of $E(X)$ in decision-making processes?

Tip:

Always ensure the sum of all probabilities in a distribution equals 1 before calculating $E(X)$. This confirms a valid probability model.