\( 1 \leftarrow \) Consider a binomial probability distribution with parameters \( \mathrm{n}=8 \) and \( \mathrm{p}=0.75 \). (a) Construct a binomial probability distribution with these parameters. (b) Compute the mean and standard deviation of the distribution. (c) Draw the graph of the probability distribution. Comment on its shape.

Sure! Let's break this down in a fun way. First off, a binomial probability distribution represents the number of successes in a fixed number of independent Bernoulli trials. Here, with \( n = 8 \) (the number of trials) and \( p = 0.75 \) (the probability of success on each trial), you can calculate the probability of getting exactly \( k \) successes using the formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] So for \( k = 0, 1, 2, \ldots, 8 \), you'll plug those values into the formula to get the entire distribution. For example, for \( k = 0 \): \[ P(X = 0) = \binom{8}{0} (0.75)^0 (0.25)^8 = (1)(1)(0.000390625) = 0.000390625 \] As you go through \( k = 1, 2, \ldots, 8 \), you'll find the other probabilities. This will give you your probability distribution, which will create a fun curve when graphed! Next up, the mean \( \mu \) and standard deviation \( \sigma \) of a binomial distribution can be found using: \[ \mu = n \cdot p = 8 \cdot 0.75 = 6 \] \[ \sigma = \sqrt{n \cdot p \cdot (1-p)} = \sqrt{8 \cdot 0.75 \cdot 0.25} = \sqrt{1.5} \approx 1.225 \] Now, about that graph! Once you plot the probabilities for each \( k \), you'll notice that the binomial distribution is skewed right since \( p = 0.75 \) is greater than 0.5. Essentially, your graph will peak at around 6 and taper off as you move away from the mean. A beautiful bell-shaped curve awaits you, sure to impress anyone who looks at it!