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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 \(P(X = x)\):

| \(X\) | 0  | 1  | 2  | 3  | 4  | 5  | 6  |
|-------|----|----|----|----|----|----|----|
| \(P(X = x)\) | 0.1 | 0.15 | 0.25 | 0.2 | 0.15 | 0.1 | 0.05 |

### (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.

Solve the given problem involving a discrete random variable X with specified probabilities.

Learn how to solve a probability distribution problem involving a discrete random variable X. This example covers key concepts such as calculating cumulative probabilities, expected value, and variance, with detailed step-by-step solutions. Suitable for students in Grades 10-12.

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