8. based upon a long period of record keeping the following represents the probability distribution of the number of times the john jay wifi network is slow during a week. we call the random variable x. x 0 1 2 3 4 5 6 p(x) .08 .17 .21 k .21 k .13 a. what is the value of k? b. calculate the expected value of x. c. calculate the expected value of x^2 . d. calculate the variance of x. e. calculate the standard deviation of x f. calculate the variance of 3x. g. suppose that network slowness is independent from week to week. what is the probability that if we look at 5 separate weeks, the network has no more than 4 slow times in any of those weeks? h. calculate the variance of the random variable x^2

Given a table representing the probability distribution of the number of times the John Jay wifi network is slow during a week. We call the random variable x.


 

Part A:

The total value of p(x) = 1.

Thus,

.08 + .17 + .21 + k + .21 + k + .13 = 1

0.8 + 2k = 1

2k = 1 - 0.8 = 0.2

k = 0.2 / 2 = 0.1

Therefore, the value of k is 0.1

Part B:

The expected value of x is given by



Therefore, the expected value of x is 3.01

Part C:

The expected value of is given by



Therefore, the expected value of is 12.45


Part D:

The variance of x is given by



Therefore, the variance of x is 3.39.

Part E

The standard deviation of x is given by



Therefore, the standard deviation of x is 1.84.

Part F:

The variance of ax, where a is a constant is given by



Thus, the variance of 3x is given by



Therefore, the variance of 3x is 30.51.

Part G:

The probability that the network has no more that 4 slow times in one week is given by



Since, the network slowness is independent from week to week, the probability that if we look at 5 separate weeks, the network has no more than 4 slow times in any of those weeks is given by



Therefore, the probability that if we look at 5 separate weeks, the network has no more than 4 slow times in any of those weeks is 0.27

Part H:

The variance of x^2 is given by





Thus,



Therefore, the variance of the random variable is 141.37