1. **Problem Statement:** You provided formulas and properties for eight different probability distributions: Binomial (two versions), Poisson, Uniform discrete, Uniform continuous, Triangular, Exponential, and Normal distributions. 2. **Focus on the first problem (a) Binomial distribution:** The probability mass function (PMF) is given by: $$p(x) = \binom{n}{x} p^x q^{n-x} \quad \text{for } x=0,1,\ldots,n$$ where $q = 1-p$. 3. **Expected value and variance:** - Expected value: $$E(x) = np$$ - Variance: $$Var(x) = npq$$ 4. **Explanation:** - The binomial distribution models the number of successes in $n$ independent Bernoulli trials, each with success probability $p$. - The PMF calculates the probability of exactly $x$ successes. - The expected value $E(x)$ is the average number of successes. - The variance $Var(x)$ measures the spread of the distribution. 5. **Example calculation:** Suppose $n=5$ trials and success probability $p=0.4$, then $q=0.6$. - Calculate $p(2)$: $$p(2) = \binom{5}{2} (0.4)^2 (0.6)^3 = 10 \times 0.16 \times 0.216 = 0.3456$$ - Expected value: $$E(x) = 5 \times 0.4 = 2$$ - Variance: $$Var(x) = 5 \times 0.4 \times 0.6 = 1.2$$ This completes the solution for the first problem.