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.

UkusakazaL

UkusakazaL

Answered question

2021-08-08

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 irs CDF.
Don't forget to specify the intervals for these functions explicitly.

Answer & Explanation

boitshupoO

boitshupoO

Beginner2021-08-09Added 1 answers

Step 1
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 :
XU(a,b)
Here, a= smallest value
b= largest value
Assume total walking time is T
i.e
T=Dx
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 =34.5=23=0.6667 and 33=1h
Total time of walking
TU(a,b)
TU(23h,1h)
The CDF of T is
F(t)=taba
=t23123
3t2
CDF can be defined as
F(t)={3t2,for23<t<1}
0, otherwise
Step 2
(b)
To calculate the PDF of T from its CDF
The PDF of T is
f(t)=F(t)
differentiate w.r.t "t"
f(t)=ddx[F(t)]
=ddt[3t2]
=30
=3
PDF of T:f(t)=3 where 23<t<1
PDF can be define as:
f(t)={3,for23<t<1}
0, otherwise

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