Question

Recently you've noticed that with many products no longer obtainable through purchase at brick and mortar locations, your family is making more purchases on various ecommerce websites.

Define X to be a random variable denoting the total number of packages delivered to your house each day. The probability mass function (pmf) of X is as follows:

x	0	1	2	3	4	5	6
P(X=x)	.18	.20	.22	.19	.13	.06	.02

What is the probability you receive zero packages back to back days?

What is the probability you receive at most two packages?

Compute the expected number of packages received.

Compute the expected value of

Compute the expected value of standard deviation of X.

Answer 1

Solution:

x	P(X)	x*P(x)	x2	x2 * P(x)
0	0.18	0	0	0
1	0.2	0.2	1	0.2
2	0.22	0.44	4	0.88
3	0.19	0.57	9	1.71
4	0.13	0.52	16	2.08
5	0.06	0.3	25	1.5
6	0.02	0.12	36	0.72
SUM	1	2.15		7.09

1)

What is the probability you receive zero packages back to back days?

Answer : 0.18

2)

What is the probability you receive at most two packages?

= P(X <= 2)

= P(X = 0) + P(X = 1) + P(X = 2)

= 0.18 + 0.2 +0.22

= 0.6

3)

Compute the expected number of packages received.

E(X)

= Summation(x.P(X))

= 2.15

E(X) = 2.15

4)

Now , E(X²) = summation [x² * P(X)] = 7.09

Variance ² = E(X²) - [E(x)]²

= 7.09 - [2.15]²

=  2.4675

Variance ² = 2.4675

5)

Standard deviation =   =  2.4675 =  1.57082780724

Standard deviation =  1.57082780724