The money spent by people at an amusement park after paying

The money spent by people at an amusement park, after paying to get in the gate, is thought to be uniformly distributed between $10.00 and $30.00.

(a) Plot the probability density function for spending at this amusement park. What is the expected value of the money spent by people?

(b) What is the probability that someone will spend between $20.00 and $25.00?

(c) Two people are selected at random. What is the probability that none of them is spending less than $15.00?

(d) A person is selected at random. This person is known to spend more than average spend- ing. What is the probability that he is also spending less than $25.00?

Solution

PDF of Uniform Distribution f(x) = 1 / ( b - a ) for a < x < b
b = Maximum Value
a = Minimum Value
Mean = a + b / 2
Standard Deviation = Sqrt ( ( b - a ) ^ 2 / 12 )

f(x) = 1/(b-a) = 1 / (30-10) = 1 / 20 = 0.05
a)
Mean = a + b / 2 = 20
Standard Deviation = Sqrt ( ( b - a ) ^ 2 / 12 ) = 5.774

b)
To find P(a < X < b) =( b - a ) * f(x)
P(20 < X < 25) = (25-20) * f(x)
= 5*0.05
= 0.25
c)
P(X < 15) = (15-10) * f(x)
= 5*0.05
= 0.25
d)
P(X > 25) = (30-25) * f(x)
= 5*0.05
= 0.25