Step-by-Step Solution for the Given Problem

We need to compute:

1. Expected value (mean) and standard deviation of $X$.
2. Expected value (mean) and standard deviation of $Y = 3X + 12$.

Given Data:

• $X = 1, 2, 3$
• $P(X=x) = 0.1, 0.35, 0.55$

(i) Expected Value of $X$:

The formula for expected value is: $E(X) = \sum_x x \cdot P(X = x)$ $E(X) = (1 \cdot 0.1) + (2 \cdot 0.35) + (3 \cdot 0.55)$ $E(X) = 0.1 + 0.7 + 1.65 = 2.45$

Thus, $E(X) = 2.45$.

(ii) Variance and Standard Deviation of $X$:

The variance formula is: $\text{Var}(X) = E(X^2) - [E(X)]^2$ First, compute $E(X^2)$: $E(X^2) = \sum_x x^2 \cdot P(X = x)$ $E(X^2) = (1^2 \cdot 0.1) + (2^2 \cdot 0.35) + (3^2 \cdot 0.55)$ $E(X^2) = (1 \cdot 0.1) + (4 \cdot 0.35) + (9 \cdot 0.55)$ $E(X^2) = 0.1 + 1.4 + 4.95 = 6.45$

Now, compute variance: $\text{Var}(X) = E(X^2) - [E(X)]^2 = 6.45 - (2.45)^2$ $\text{Var}(X) = 6.45 - 6.0025 = 0.4475$

Standard deviation: $\text{SD}(X) = \sqrt{\text{Var}(X)} = \sqrt{0.4475} \approx 0.67$

Thus, $\text{SD}(X) \approx 0.67$.

(iii) Expected Value of $Y = 3X + 12$:

The expected value of $Y$ is: $E(Y) = E(3X + 12) = 3E(X) + 12$ $E(Y) = 3(2.45) + 12 = 7.35 + 12 = 19.35$

Thus, $E(Y) = 19.35$.

(iv) Variance and Standard Deviation of $Y$:

The variance of $Y$ is: $\text{Var}(Y) = \text{Var}(3X + 12) = 3^2 \cdot \text{Var}(X)$ $\text{Var}(Y) = 9 \cdot 0.4475 = 4.0275$

Standard deviation of $Y$: $\text{SD}(Y) = \sqrt{\text{Var}(Y)} = \sqrt{4.0275} \approx 2.01$

Thus, $\text{SD}(Y) \approx 2.01$.

Final Results:

1. $E(X) = 2.45$, $\text{SD}(X) \approx 0.67$
2. $E(Y) = 19.35$, $\text{SD}(Y) \approx 2.01$

Would you like additional details or further clarification?

Follow-Up Questions:

1. What are the implications of these values in a real-world context?
2. How does the transformation $Y = 3X + 12$ affect the variance and standard deviation?
3. What if the probabilities $P(X=x)$ were changed? How would the computations differ?
4. Can this approach be generalized to more complex probability distributions?
5. How can this analysis be visualized graphically?

Tip:

Always check the sum of probabilities $P(X=x)$ to ensure they equal 1 before proceeding with calculations.