The problem you provided seems to be about finding specific properties related to a probability distribution function $f(x)$. Let's go step by step through the questions listed in the image.

The first part of the image is:

1. Find $A$: This refers to determining the constant $A$ that makes the probability function $f(x)$ a valid probability distribution. The function is given as:

   0 & \text{for } x \leq -4 \\ A(x+4) & \text{for } -4 < x \leq 1 \\ 0 & \text{for } x > 1 \end{cases}$$ To ensure this is a valid probability density function, it must satisfy the condition that the total probability integrates to 1. So, we need to solve the integral of $$f(x)$$ over the range where $$f(x)$$ is non-zero, from $$-4$$ to $$1$$: $$\int_{-4}^{1} A(x + 4) \, dx = 1$$ Let's solve this integral.

2. Compute $P(0 < X < 1)$: This asks to find the probability that the random variable $X$ lies between 0 and 1, which can be calculated as the integral of $f(x)$ from 0 to 1: $P(0 < X < 1) = \int_0^1 A(x + 4) \, dx$

3. Find the cumulative distribution function $F(x)$: The cumulative distribution function is given by the integral of the probability density function: $F(x) = \int_{-\infty}^{x} f(t) \, dt$

4. Compute the expected value $M(X)$ and variance $D(X)$: The expected value and variance can be calculated using: $M(X) = \int_{-\infty}^{\infty} x f(x) \, dx$ and $D(X) = \int_{-\infty}^{\infty} (x - M(X))^2 f(x) \, dx$

5. Plot the graphs of $f(x)$ and $F(x)$: This would require plotting the functions based on the given information.

I'll begin by solving for $A$ first. sympy import symbols, integrate, Eq, solve

Define the variable

x, A = symbols('x A')

Define the probability density function

f_x = A * (x + 4)

Set up the integral to find A

integral_eq = Eq(integrate(f_x, (x, -4, 1)), 1)

Solve for A

A_value = solve(integral_eq, A) A_valueThe value of $A$ is $\frac{2}{25}$.

Now that we know $A$, we can proceed with the second part, which is to compute the probability $P(0 < X < 1)$.

Let's compute this integral using the value of $A$. Substitute A value into the function and compute the probability P(0 < X < 1) A_value = A_value[0] probability_integral = integrate(A_value * (x + 4), (x, 0, 1)) probability_integralThe probability $P(0 < X < 1)$ is $\frac{9}{25}$.

Would you like me to proceed with the next steps, such as calculating the cumulative distribution function and the expected value/variance? Let me know if you'd like more details!