To solve this problem, we need to calculate the mean ($\mu$) and the standard deviation ($\sigma$) of the random variable $X$, based on the given probability distribution.

Step (a): Find the Mean ($\mu$)

The mean of a discrete random variable is calculated using the formula: $\mu = \sum_{x} x \cdot P(X=x)$ From the table:

• $X = 1, 2, 3$
• Probabilities $P(X=1) = 0.4$, $P(X=2) = 0.4$, $P(X=3) = 0.2$

$\mu = (1 \cdot 0.4) + (2 \cdot 0.4) + (3 \cdot 0.2)$ $\mu = 0.4 + 0.8 + 0.6 = 1.8$

Step (b): Find the Standard Deviation ($\sigma$)

The formula for standard deviation is: $\sigma = \sqrt{\sum_{x} (x - \mu)^2 \cdot P(X=x)}$

First, calculate $(x - \mu)^2 \cdot P(X=x)$ for each $x$:

• For $X=1$: $(1 - 1.8)^2 \cdot 0.4 = (-0.8)^2 \cdot 0.4 = 0.64 \cdot 0.4 = 0.256$
• For $X=2$: $(2 - 1.8)^2 \cdot 0.4 = (0.2)^2 \cdot 0.4 = 0.04 \cdot 0.4 = 0.016$
• For $X=3$: $(3 - 1.8)^2 \cdot 0.2 = (1.2)^2 \cdot 0.2 = 1.44 \cdot 0.2 = 0.288$

Now, sum these values: $\text{Variance } (\sigma^2) = 0.256 + 0.016 + 0.288 = 0.56$

Take the square root to find the standard deviation: $\sigma = \sqrt{0.56} \approx 0.75$

Final Answers:

• $\mu = 1.8$ (rounded to 2 decimal places)
• $\sigma = 0.75$ (rounded to 2 decimal places)

Would you like further details or explanations?

Here are 5 related questions for practice:

1. What is the variance of this random variable?
2. How would the mean change if $P(X=3)$ increased to 0.3?
3. How does standard deviation differ conceptually from variance?
4. What is the cumulative distribution function (CDF) of $X$?
5. If another random variable $Y$ has the same mean but a higher standard deviation, what does that imply about $Y$?

Tip: Always verify that the sum of probabilities $\sum P(X=x)$ equals 1 to ensure the distribution is valid.