The internet connection speed at any time from your home can depend on the amoun

Question

The internet connection speed at any time from your home can depend on the amount of overall internet trac at that time. Let the random variable X denote the speed of connection in megabits per second (MBPS). Assuming X is uniformly distributed on the interval 0:75 to 1:25 MBPS, answer the following questions: (a) Find the mean connection speed and standard deviation. (b) What is the probability that the connection speed will be less than 0.8 MBPS at any given time? (c) What is the probability that the connection speed will be between 0.875 MBPS and 1.125 MBPS at any given time?

Explanation / Answer

X follows uniform (0.75,1.25),
2X-1.5 follows Uniform(0,1)
a) E(2X-1.5) = 0.5 ==> E(X) = 2/2 =1
since Y = 2X-1.5 has variance 1/3 and variance (Y) = 4* variance (X)
variance (X) =1/12 ==>and std deviation of X = sqrt(1/12)
b)P(X<0.8) = (2X-1.5 < 0.1) =(Y<0.1) = 0.1
c) P(0.875<X<1.125) = P(2(0.875)-1.5< 2X-1.5 < 2(1.125)-1.5) = P(0.25 < Y < 0.75) = 0.75 - 0.25= 0.5

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