We walk from Central Park to Empire State Building with a constant speed. Our speed X is a random variable uniformly distributed between 3 and 4.5 km/h. The distance is 3 km. Calculate the probability distribution of the total walking time T: a. Calculate the CDF of T from the distribution of X. b. Calculate the PDF of T from its CDF. Don't forget to specify the intervals for these functions explicitly.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Speed is uniformly distributed between 3 and 4.5 km/h from central park to empire state.
(a)
To calculate the CDF of T from the distribution of X.
The random variable X for the speed is given by :
X U ( a,b )
Here, a = smallest value
b = largest value
Assume total walking time is T
i.e
T =
As Distance D = 3km is uniformly distributed.
Time is inversely proportional to the speed, when the speed is maximum and time is minimum and vice - versa
Time = and = 1h
Total time of walking
T U ( a , b )
T U ( h , 1h )
The CDF of T is
F(t) =
CDF can be defined as
F (t) =
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