Given the probability distribution of the random variable $X$, we can calculate the expectation $E(X)$ and the variance $\text{Var}(X)$.

Probability Distribution:

Value $x$ of $X$	$P(X = x)$
2	0.15
1	0.20
0	0.10
1	0.10
2	0.45

However, notice that the values of $X$ have repeated probabilities. To simplify, we should group the same values of $X$ together:

• For $X = 2$: $P(X = 2) = 0.15 + 0.45 = 0.60$
• For $X = 1$: $P(X = 1) = 0.20 + 0.10 = 0.30$
• For $X = 0$: $P(X = 0) = 0.10$

Simplified Probability Distribution:

Value $x$ of $X$	$P(X = x)$
2	0.60
1	0.30
0	0.10

(a) Expectation $E(X)$:

The expectation $E(X)$ is calculated using the formula: $E(X) = \sum_{i} x_i \cdot P(X = x_i)$ Substituting the values: $E(X) = (2 \times 0.60) + (1 \times 0.30) + (0 \times 0.10) = 1.20 + 0.30 + 0 = 1.50$ So, $E(X) = 1.50$.

(b) Variance $\text{Var}(X)$:

The variance $\text{Var}(X)$ is calculated using the formula: $\text{Var}(X) = E(X^2) - [E(X)]^2$ First, we calculate $E(X^2)$: $E(X^2) = \sum_{i} x_i^2 \cdot P(X = x_i)$ Substituting the values: $E(X^2) = (2^2 \times 0.60) + (1^2 \times 0.30) + (0^2 \times 0.10) = (4 \times 0.60) + (1 \times 0.30) + (0 \times 0.10) = 2.40 + 0.30 + 0 = 2.70$

Now, using $E(X) = 1.50$: $\text{Var}(X) = E(X^2) - [E(X)]^2 = 2.70 - (1.50)^2 = 2.70 - 2.25 = 0.45$

Final Answers:

• $E(X) = 1.50$
• $\text{Var}(X) = 0.45$

Would you like more details or have any questions?

Related Questions:

1. How do you interpret the expectation $E(X)$ in the context of a probability distribution?
2. How does the variance $\text{Var}(X)$ reflect the spread of the distribution?
3. What is the standard deviation of $X$ given the variance $\text{Var}(X)$?
4. How would the expectation change if $P(X = 0)$ were increased to 0.20?
5. What is the cumulative distribution function (CDF) for this probability distribution?
6. How do you compute $E(X^3)$ for this distribution?
7. What would happen to the variance if all probabilities were uniformly distributed?
8. How does variance relate to the concept of risk in probability?

Tip:

When calculating expectation and variance, always check for any repeated values in the distribution to simplify the calculations.