Use $n=6$ and $p=0.25$ to complete parts (a) through (d) below. (a) Construct a binomial probability distribution with the given parameters. \begin{tabular}{|c|c|} \hline$x$ & $P(x)$ \\ \hline 0 & .178 \\ \hline 1 & 356 \\ \hline 2 & 2966 \\ \hline 3 & 1318 \\ \hline 4 & .033 \\ \hline 5 & .0044 \\ \hline 6 & 0002 \\ \hline \end{tabular} (Round to four decimal places as needed.) (b) Compute the mean and standard deviation of the random variable using $\mu_{x}=\sum[x \cdot P(x)]$ and $\sigma_{x}=\sqrt{\sum\left[x^{2} \cdot P(x)\right]-\mu_{x}^{2}}$. $\mu_{x}=\square$ (Round to two decimal places as needed)

Step 1 ：The binomial probability distribution is given by the formula: \(P(x) = C(n, x) * (p^x) * ((1-p)^{n-x})\), where \(C(n, x)\) is the combination of n items taken x at a time, p is the probability of success, and \((1-p)\) is the probability of failure.

Step 2 ：Given \(n=6\) and \(p=0.25\), we can calculate \(P(x)\) for \(x=0\) to \(6\).

Step 3 ：\(P(0) = C(6, 0) * (0.25^0) * ((1-0.25)^{6-0}) = 1 * 1 * 0.1779785 = 0.1779785\)

Step 4 ：\(P(1) = C(6, 1) * (0.25^1) * ((1-0.25)^{6-1}) = 6 * 0.25 * 0.2373046875 = 0.35671875\)

Step 5 ：\(P(2) = C(6, 2) * (0.25^2) * ((1-0.25)^{6-2}) = 15 * 0.0625 * 0.31640625 = 0.296630859375\)

Step 6 ：\(P(3) = C(6, 3) * (0.25^3) * ((1-0.25)^{6-3}) = 20 * 0.015625 * 0.421875 = 0.1318359375\)

Step 7 ：\(P(4) = C(6, 4) * (0.25^4) * ((1-0.25)^{6-4}) = 15 * 0.00390625 * 0.5625 = 0.033203125\)

Step 8 ：\(P(5) = C(6, 5) * (0.25^5) * ((1-0.25)^{6-5}) = 6 * 0.0009765625 * 0.75 = 0.00439453125\)

Step 9 ：\(P(6) = C(6, 6) * (0.25^6) * ((1-0.25)^{6-6}) = 1 * 0.000244140625 * 1 = 0.000244140625\)

Step 10 ：The mean and standard deviation of the random variable can be calculated using the formulas: Mean, \(\mu_x = \Sigma[x * P(x)]\) and Standard deviation, \(\sigma_x = \sqrt{\Sigma[x^2 * P(x)] - \mu_x^2}\)

Step 11 ：Mean, \(\mu_x = 0*P(0) + 1*P(1) + 2*P(2) + 3*P(3) + 4*P(4) + 5*P(5) + 6*P(6) = 1.5\)

Step 12 ：Standard deviation, \(\sigma_x = \sqrt{[0^2*P(0) + 1^2*P(1) + 2^2*P(2) + 3^2*P(3) + 4^2*P(4) + 5^2*P(5) + 6^2*P(6)] - 1.5^2} = 1.12\)

Step 13 ：So, the mean is \(\boxed{1.5}\) and the standard deviation is \(\boxed{1.12}\)