Question

1. suit sales the number of suits sold per day at a retail store is shown in the table, with the corresponding probabilities. find the mean, variance, and standard deviation of the distribution.

number of suits sold x | 19 20 21 22 23
probability p(x) | 0.2 0.2 0.3 0.2 0.1
if the manager of the retail store wants to be sure that he has enough suits for the next 5 days, how many should the manager purchase?

Explanation:

Step1: Calculate the mean ($\mu$)

The formula for the mean of a discrete - probability distribution is $\mu=\sum xP(x)$.
$\mu=(19\times0.2)+(20\times0.2)+(21\times0.3)+(22\times0.2)+(23\times0.1)$
$=3.8 + 4+6.3 + 4.4+2.3$
$=20.8$

Step2: Calculate the variance ($\sigma^{2}$)

The formula for the variance is $\sigma^{2}=\sum(x - \mu)^{2}P(x)$.
$(19 - 20.8)^{2}\times0.2+(20 - 20.8)^{2}\times0.2+(21 - 20.8)^{2}\times0.3+(22 - 20.8)^{2}\times0.2+(23 - 20.8)^{2}\times0.1$
$=(-1.8)^{2}\times0.2+(-0.8)^{2}\times0.2+(0.2)^{2}\times0.3+(1.2)^{2}\times0.2+(2.2)^{2}\times0.1$
$=(3.24\times0.2)+(0.64\times0.2)+(0.04\times0.3)+(1.44\times0.2)+(4.84\times0.1)$
$=0.648+0.128 + 0.012+0.288+0.484$
$=1.56$

Step3: Calculate the standard deviation ($\sigma$)

The standard deviation is the square - root of the variance, $\sigma=\sqrt{\sigma^{2}}$.
$\sigma=\sqrt{1.56}\approx1.25$

Step4: Determine the number of suits for 5 days

The expected number of suits sold per day is $\mu = 20.8$. For 5 days, the manager should purchase $20.8\times5=104$ suits. Since we can't have a fraction of a suit in practice, the manager should purchase 104 suits.

Answer:

Mean: $20.8$, Variance: $1.56$, Standard Deviation: approximately $1.25$, Number of suits to purchase for 5 days: 104