The image shows a problem involving a discrete random variable $X$ with given probabilities. Here's the step-by-step solution to the questions presented:

Given Data:

The table presents the values of $X$ and their corresponding probabilities $P(X = x)$:

(a) Probability that $X$ is at most 3.

This means finding $P(X \leq 3)$:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

$P(X \leq 3) = 0.1 + 0.15 + 0.25 + 0.2 = 0.7$

(b) Probability that $X$ is greater than 3.

This means finding $P(X > 3)$:

$P(X > 3) = P(X = 4) + P(X = 5) + P(X = 6)$

$P(X > 3) = 0.15 + 0.1 + 0.05 = 0.3$

(c) Probability that $X$ is an even number.

Even values of $X$ are $0, 2, 4, 6$. Thus, $P(X$ is even$) = P(X = 0) + P(X = 2) + P(X = 4) + P(X = 6)$:

$P(X \text{ is even}) = 0.1 + 0.25 + 0.15 + 0.05 = 0.55$

(d) Mean (Expected Value), $E(X)$.

The expected value is calculated as:

$E(X) = \sum X \times P(X)$

$E(X) = (0 \times 0.1) + (1 \times 0.15) + (2 \times 0.25) + (3 \times 0.2) + (4 \times 0.15) + (5 \times 0.1) + (6 \times 0.05)$

$E(X) = 0 + 0.15 + 0.5 + 0.6 + 0.6 + 0.5 + 0.3 = 2.65$

(e) Variance, $Var(X)$.

Variance is calculated using:

$Var(X) = E(X^2) - (E(X))^2$

First, calculate $E(X^2)$:

$E(X^2) = (0^2 \times 0.1) + (1^2 \times 0.15) + (2^2 \times 0.25) + (3^2 \times 0.2) + (4^2 \times 0.15) + (5^2 \times 0.1) + (6^2 \times 0.05)$

$E(X^2) = 0 + 0.15 + 1 + 1.8 + 2.4 + 2.5 + 1.8 = 9.65$

Now, use the expected value found earlier:

$Var(X) = 9.65 - (2.65)^2 = 9.65 - 7.0225 = 2.6275$

(f) Probability that $0.5 \leq X \leq 4.5$.

This range includes $X = 1, 2, 3, 4$:

$P(0.5 \leq X \leq 4.5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$

$P(0.5 \leq X \leq 4.5) = 0.15 + 0.25 + 0.2 + 0.15 = 0.75$

Let me know if you need further details or have any questions!

Related Questions:

1. What would be the probability that $X$ is less than 2?
2. How would you calculate the standard deviation of $X$?
3. What is the probability that $X$ is an odd number?
4. How do you interpret the expected value in a probability distribution?
5. How would the variance change if all probabilities were doubled?

Tip: Always double-check your calculations when working with probability distributions, especially when summing probabilities and computing expected values or variances.