Question

A mail-order computer business has six telephone lines. Let X denote the number of lines in use at a specified time. Suppose the pmf of X is given in the accompanying table
X 0 1 2 3 4 5 6
p(x) .04 .21 .25 .30 .10 .08 .02
Calculate the probability of each of the following events:
(a) At most 4 lines are in use.
(b) Fewer than 2 lines are in use.
(c) At least 3 lines are in use.
2. If k is a constant and X and Y are random variables, then
(i) E(k) = ? (ii) E(kX) = ? (iii) E(X ± Y) = ?

Answer 1

Answer:

a. 0.9

b. 0.25

c. 0.5

i. k

ii. k..E( X )

iii. E ( X ) ± E ( Y )

Step-by-step explanation:

Solution:-

- The probability distribution of a random variable X that denotes the number of lines used in a period of time t. The probability mass function of the RV X is given below:

X                   0            1               2              3              4             5              6

p ( X )          0.04      0.21         0.25         0.30          0.1         0.08         0.02

- We are to calculate the following probabilities:

a) At most four lines are in use.

- Translate the required probability in mathematical expression as follows:

          p ( X ≤ 4 ) = p ( X = 0 ) + p ( X = 1 ) + p ( X = 2 ) + p ( X = 3 ) + p ( X = 4 )

- Use the given pmf for the random variable X and determine the required probability as follows:

         p ( X ≤ 4 ) = 0.04 + 0.21 + 0.25 + 0.3 + 0.1

         p ( X ≤ 4 ) = 0.9  ... Answer

b) Fewer than two lines are in use.

- Translate the required probability in mathematical expression as follows:

          p ( X < 2 ) = p ( X = 0 ) + p ( X = 1 )

- Use the given pmf for the random variable X and determine the required probability as follows:

         p ( X < 2 ) = 0.04 + 0.21

         p ( X < 2 ) = 0.25  ... Answer

c) At least three lines are in use.

- Translate the required probability in mathematical expression as follows:

          p ( X ≥ 3 ) =  p ( X = 3 ) + p ( X = 4 ) + p ( X = 5 ) + p ( X = 6 )

- Use the given pmf for the random variable X and determine the required probability as follows:

         p ( X ≥ 3 ) =  0.3 + 0.1 +0.08 + 0.02

         p ( X ≥ 3 ) = 0.5  ... Answer

2. The following properties of the expected value of random variables are as follows:

        E ( k ) = k ... provided k is a constant for all iterative values

       E ( k.X ) = ∑ $( k.X_i.p( X_i ) )$ = k*∑ $( X_i.p( X_i ) )$ = k. E ( X )

- For two random variables X and Y the arithmetic expected value is defined as:

      E ( X ± Y ) = E ( X ) ± E ( Y ) ... only if both E ( X ) and E ( Y ) exist.