Consider the following probability distribution function of the random variable X which represents the number of people in a group (party) at a restaurant: X P(X) 1 0.10 2 0.25 3 0.20 4 0.20 5 0.10 6 7 0.05 8 0.05 a. Find the population mean of X. b. Find the population variance and standard deviation of X. c. Find the probability that the next party will be over 4 people. d. Find the probability that the next three parties (assuming independence) will each be over 4 people.

(a) 3.55

(b) 3.45 and 1.86

(c) 0.25

(d) 0.016

Step-by-step explanation:

The random variable X 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:

Consider the Excel sheet attached.

Compute the variance as follows:

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.

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btw in my country we just have sun and rain as (because i live in the caribbean lol)

also, thx a lot for the points : )