Probability density function for x which is uniformly distributed [a,b] :
[tex]F(x)=\dfrac{1}{b-a}[/tex]
Given : items showed an approximately uniform distribution between 20 minutes and 40 minutes.
Let x be the time spent shopping in a supermarket.
Probability density function for x : [tex]f(x)=\dfrac{1}{40-20}=0.05[/tex]
a.
[tex]P(25<x<30)=\int^{30}_{25}f(x)\ dx\\\\ =0.05\int^{30}_{25}\ dx\\\\=0.05[x]^{30}_{25}\\\\= 0.05(30-25)=0.25[/tex]
∴The probability that the shopping time will be between 25 and 30 minutes is 0.25.
b.
[tex]P(x<35)=\int^{35}_{25}f(x)\ dx\\\\ =0.05\int^{35}_{25}\ dx\\\\=0.05[x]^{35}_{25}\\\\= 0.05(35-25)=0.5[/tex]
∴The probability that the shopping time will be less than 35minutes is 0.5
c. Mean : [tex]\mu=\dfrac{a+b}{2}=\dfrac{20+40}{2} =30 \text{ minutes}[/tex]
Standard deviations : [tex]\sigma=\sqrt{\dfrac{(b-a)^2}{12}}=\sqrt{\dfrac{(40-20)^2}{12}}\approx5.77\text{ minutes}[/tex]
i.e. the mean and standard deviation of the shopping time are 30 minutes and 5.77 minutes.
