Question

The BRIT Sports Bar sells a large quantity of Guinness every Saturday. From past records, the pub has determined the following probabilities for sales:

 

 

  Number of Crates (X) P(x)

18 0.15

19 0.10

20 0.32

21 0.05

22 0.13

23 0.25

 

a.   Verify that this [P(x)] is a probability distribution.                              

e.   Calculate the variance of the distribution?                                         

f.    Determine the standard deviation of the distribution?                         

 

Answer 1

a. To verify P(X) is a probability function we have to show that $\sum_{X} P(X) = 1$.

$\begin{aligned} \sum_{x} P(x) &= P(18) + P(19) + P(20) + P(21)+P(22)+P(23)\\ &= 0.15+0.10+0.32+0.05+0.13+0.25\\ &=1 \end{aligned}$

Hence P(X) is a probability function.

e. $\text{Variance}(X) = E(X^2) - (E(X))^2$ .

$\begin{aligned} E(X^2) &= \sum_{x} X^2 P(X)\\ & = 18^2*0.15+19^2*0.10+20^2*0.32~+~\\ & ~~~~~~~~~~~~~21^2*0.05+22^2*0.13+23^2*0.25\\ &=429.92 \end{aligned}$

$\begin{aligned} E(X) &= \sum_{x} X P(X)\\ & = 18*0.15+19*0.10+20*0.32~+~\\ & ~~~~~~~~~~~~~21*0.05+22*0.13+23*0.25\\ &=20.66 \end{aligned}$

$\therefore \text{Variance}(X) = 429.92-20.66^2 = 429.92-426.8356 = 3.0844$

f. $\text{Standard deviation}(X) = \sqrt{\text{Var}(X)} = \sqrt{3.0844} = 1.7562$