Answer:
(a) 3.55
(b) 3.45 and 1.86
(c) 0.25
(d) 0.016
Step-by-step explanation:
The random variable <em>X</em> denotes the number of people in a group (party) at a restaurant.
(a)
The formula to compute the mean is:

Consider the Excel sheet attached.
The mean is, 3.55.
(b)
The formula to compute the variance is:
![\sigma^{2}=[\sum {x^{2}\cdot P(X=x)}]-(\mu)^{2}](https://tex.z-dn.net/?f=%5Csigma%5E%7B2%7D%3D%5B%5Csum%20%7Bx%5E%7B2%7D%5Ccdot%20P%28X%3Dx%29%7D%5D-%28%5Cmu%29%5E%7B2%7D)
Consider the Excel sheet attached.
Compute the variance as follows:
![\sigma^{2}=[\sum {x^{2}\cdot P(X=x)}]-(\mu)^{2}\\\\=16.05-(3.55)^{2}\\\\=3.4475\\\\\approx 3.45](https://tex.z-dn.net/?f=%5Csigma%5E%7B2%7D%3D%5B%5Csum%20%7Bx%5E%7B2%7D%5Ccdot%20P%28X%3Dx%29%7D%5D-%28%5Cmu%29%5E%7B2%7D%5C%5C%5C%5C%3D16.05-%283.55%29%5E%7B2%7D%5C%5C%5C%5C%3D3.4475%5C%5C%5C%5C%5Capprox%203.45)
The variance is, 3.45.
Compute the standard deviation as follows:

The standard deviation is, 1.86.
(c)
Compute the probability that the next party will be over 4 people as follows:

Thus, the probability that the next party will be over 4 people is 0.25.
(d)
Compute the probability that the next three parties will each be over 4 people as follows:
It is provided that the three parties are independent.
P (Next 3 parties will be each over 4) = [P (X > 4)]³

Thus, the probability that the next three parties will each be over 4 people is 0.016.