To find the mean (or expected value) of the distribution given by the function $f(x) = \frac{3}{4}(x - a)(b - x)$ on the interval $[a, b]$, where $a = 1$ and $b = 3$, we can follow these steps:

1. Normalizing the distribution:

We need to ensure that the function $f(x)$ is normalized, i.e., the total integral over the interval $[a, b]$ is 1. Let's check if this is the case.

$\int_{1}^{3} f(x) \, dx = 1$

Given that $f(x) = \frac{3}{4}(x - 1)(3 - x)$, we can now compute this integral and check.

2. Formula for the mean:

The mean (expected value) of a probability distribution is given by:

$\mu = \int_{a}^{b} x f(x) \, dx$

Here, that becomes:

$\mu = \int_{1}^{3} x \cdot \frac{3}{4}(x - 1)(3 - x) \, dx$

3. Expand the function:

We can first expand $(x - 1)(3 - x)$ to make the integration easier:

$(x - 1)(3 - x) = 3 - x - 3x + x^2 = x^2 - 4x + 3$

Thus, the function $f(x)$ becomes:

$f(x) = \frac{3}{4}(x^2 - 4x + 3)$

Now, we substitute this into the formula for the mean:

$\mu = \int_{1}^{3} x \cdot \frac{3}{4}(x^2 - 4x + 3) \, dx$

4. Simplify the integral:

Distribute $x$ inside the integral:

$\mu = \frac{3}{4} \int_{1}^{3} (x^3 - 4x^2 + 3x) \, dx$

Now, we can integrate each term separately:

$\mu = \frac{3}{4} \left( \frac{x^4}{4} - \frac{4x^3}{3} + \frac{3x^2}{2} \right) \Bigg|_{1}^{3}$

5. Compute the integrals:

Now, calculate the values of these integrals at $x = 3$ and $x = 1$:

• At $x = 3$:

$\frac{(3^4)}{4} - \frac{4(3^3)}{3} + \frac{3(3^2)}{2} = \frac{81}{4} - 36 + \frac{27}{2}$

• At $x = 1$:

$\frac{1^4}{4} - \frac{4(1^3)}{3} + \frac{3(1^2)}{2} = \frac{1}{4} - \frac{4}{3} + \frac{3}{2}$

Now subtract the results from $x = 3$ and $x = 1$, then multiply by $\frac{3}{4}$ to find the mean.

Would you like me to continue with the detailed evaluation of the integrals, or do you want to try it on your own?

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Here are 5 questions to deepen your understanding:

1. How would the mean change if $a$ and $b$ were different values?
2. What happens to the probability distribution if you modify the normalization factor?
3. Can you derive the median of this distribution, and how does it compare to the mean?
4. How does the choice of the interval $[a, b]$ affect the shape of the function $f(x)$?
5. Can you calculate the variance of this distribution using the same method?

Tip: Integrating polynomial terms is easier if you break it down step by step. Double-check your bounds and remember to apply the fundamental theorem of calculus.