Question

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.

Answer 1

Answer:

(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:

$\mu=\sum {x\cdot P(X=x)}$

Consider the Excel sheet attached.

The mean is, 3.55.

(b)

The formula to compute the variance is:

$\sigma^{2}=[\sum {x^{2}\cdot P(X=x)}]-(\mu)^{2}$

Consider the Excel sheet attached.

Compute the variance as follows:

$\sigma^{2}=[\sum {x^{2}\cdot P(X=x)}]-(\mu)^{2}\\\\=16.05-(3.55)^{2}\\\\=3.4475\\\\\approx 3.45$

The variance is, 3.45.

Compute the standard deviation as follows:

$\sigma=\sqrt{\sigma^{2}}\\\\=\sqrt{3.45}\\\\=1.85742\\\\\approx 1.86$

The standard deviation is, 1.86.

(c)

Compute the probability that the next party will be over 4 people as follows:

$P(X>4)=P(X=5)+P(X=6)+P(X=7)+P(X=8)\\\\=0.10+0.05+0.05+0.05\\\\=0.25$

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

                                                           $=(0.25)^{3}\\=0.015625\\\approx 0.016$

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