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The uploaded image contains assignments on "Probability and Random Processes." I will proceed by analyzing the problems and providing solutions step by step.

### Problem P.1

The problem describes a probability density function (pdf) \( f_X(x) \) as follows:

\[
f_X(x) = 
\begin{cases}
    Cx^2 & \text{if } 0 \leq x \leq 1 \\
    C(2 - x) & \text{if } 1 < x \leq 2 \\
    0 & \text{otherwise}
\end{cases}
\]

**Questions:**
1. (a) Find the constant \( C \).
2. (b) Find \( \Pr(0.75 \leq X \leq 1.5) \).
3. (c) Find the value of \( \alpha \) such that \( \Pr(X \leq \alpha) = 0.9 \).

### Solution to P.1

**(a) Finding \( C \)**

To find the constant \( C \), we need to use the property that the total probability must be 1:

\[
\int_{-\infty}^{\infty} f_X(x) dx = 1.
\]

Since \( f_X(x) \) is non-zero only from 0 to 2, we have:

\[
\int_0^1 Cx^2 dx + \int_1^2 C(2 - x) dx = 1.
\]

Calculating each integral:
\[
\int_0^1 Cx^2 dx = C \int_0^1 x^2 dx = C \left[ \frac{x^3}{3} \right]_0^1 = \frac{C}{3},
\]
\[
\int_1^2 C(2 - x) dx = C \int_1^2 (2 - x) dx = C \left[ 2x - \frac{x^2}{2} \right]_1^2 = C \left( 4 - 2 - (2 - 0.5) \right) = \frac{C}{2}.
\]

Adding them:
\[
\frac{C}{3} + \frac{C}{2} = 1 \Rightarrow \frac{2C}{6} + \frac{3C}{6} = 1 \Rightarrow \frac{5C}{6} = 1 \Rightarrow C = \frac{6}{5}.
\]

**(b) Finding \( \Pr(0.75 \leq X \leq 1.5) \)**

\[
\Pr(0.75 \leq X \leq 1.5) = \int_{0.75}^1 \frac{6}{5} x^2 dx + \int_1^{1.5} \frac{6}{5} (2 - x) dx.
\]

Calculate each integral separately:
\[
\int_{0.75}^1 \frac{6}{5} x^2 dx = \frac{6}{5} \left[ \frac{x^3}{3} \right]_{0.75}^1 = \frac{6}{15} (1 - 0.421875) = 0.23125,
\]
\[
\int_1^{1.5} \frac{6}{5} (2 - x) dx = \frac{6}{5} \left[ 2x - \frac{x^2}{2} \right]_1^{1.5} = 0.27.
\]

Adding both:
\[
\Pr(0.75 \leq X \leq 1.5) = 0.23125 + 0.27 = 0.50125.
\]

**(c) Finding \( \alpha \) such that \( \Pr(X \leq \alpha) = 0.9 \)**

To find \( \alpha \), we solve:

\[
\int_0^{\alpha} f_X(x) dx = 0.9.
\]

### Problem P.2

A Gaussian random variable \( x \) is with mean \( \mu = 1 \) and variance \( \sigma^2 = 3 \). Find:

1. **(a)** \( \Pr(X > \sqrt{3}) \) and \( \Pr(X < 1) \)
2. **(b)** \( \Pr(|X - 1| > 12) \) and \( \Pr(|X - 1| < 6) \)
3. **(c)** \( \Pr(X > 1 + \sqrt{3}) \)

### P.2 Solution 

Follow to find:
1. **Tip:** Use Gaussian table. 

**Need Clarifications?** Let me know.

Assignments on probability and random processes with two main problems: (P.1) a random variable X with a given probability density function (pdf) and (P.2) a Gaussian random variable and other given conditions.

This problem involves determining the constant for a given probability density function and finding specific probabilities, including Gaussian distribution calculations. Suitable for university-level students studying probability theory and random processes.

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