Let's go through the questions one by one:

Question 4:

A variable $X$ is distributed randomly between the values 0 and 4, with the probability density function (PDF) given by:

$f(x) = kx^2(4 - x^2)$

To find the value of $k$, we must use the fact that the total probability for a PDF is 1, meaning:

$\int_{0}^{4} f(x) \, dx = 1$

Substituting the given PDF:

$\int_{0}^{4} kx^2(4 - x^2) \, dx = 1$

This will give us the value of $k$.

Question 5:

In four tosses of a coin, let $X$ be the number of heads. You are asked to:

• Tabulate the 16 possible outcomes and the corresponding values of $X$.
• Find the probability distribution of $X$, the expected value, and the variance.

Here, $X$ follows a binomial distribution with parameters:

• $n = 4$ (number of tosses),
• $p = 0.5$ (probability of getting heads in each toss).

We can calculate the probabilities for each value of $X = 0, 1, 2, 3, 4$ using the binomial formula: $P(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}$

After determining the distribution, we can compute the expected value and variance:

• Expected value $E[X] = np$,
• Variance $\text{Var}(X) = np(1 - p)$.

Question 6:

$X$ is the number scored in a throw of a fair die. The task involves applying Chebyshev's inequality and finding the actual probability.

Chebyshev's inequality gives the probability that $X$ deviates from its mean $\mu$ by more than 2.5: $P(|X - \mu| > 2.5) < 0.47$

For a fair die, $X$ takes values from 1 to 6, so the mean $\mu$ is: $\mu = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$

We'll also find the actual probability using the distribution of $X$.

Would you like me to solve one of these in more detail? Here are 5 related questions to deepen your understanding:

1. How do you calculate the variance of a continuous random variable using a PDF?
2. What is the interpretation of Chebyshev's inequality in practical terms?
3. How does binomial distribution relate to the probability of heads in multiple coin tosses?
4. What are the steps to compute the actual probability in question 6?
5. How would the expected value change if the coin in question 5 was biased?

Tip: Always verify the normalization condition when dealing with probability density functions (i.e., the total probability should integrate to 1).