Q3: 3) A museum has free admission, but suggests that patrons make a donation to

Question

Q3: 3) A museum has free admission, but suggests that patrons make a donation to support it.Let X be a random variable representing the amount of money placed in the money jar by patrons who contribute.(We are ignoring all the people who do not contribute for the moment.)The probability distribution for X is as follows: P(X = $1.00) = 1/4; P(X = $2.00) = 1/6; P(X = $3.00) = 1/6; P(X = $4.00) = 1/6; P(X = $5.00) = 1/4.

(a) How much money does the museum get, on average, from any individual person who makes a donation?   What is the shape of distribution of outcomes based on a single person?

(b) Let’s suppose that 100 people per hour make a donation. How much money would the museum get, on average, each hour from donations? If you had a data set with the hourly donations sampled across many hours, what shape would the distribution have?

(C) Use the code from the class activities to make a histogram of simulated hourly revenue based on (b) above.

Explanation / Answer

The probability distribution for X is as follows: P(X = $1.00) = 1/4; P(X = $2.00) = 1/6; P(X = $3.00) = 1/6; P(X = $4.00) = 1/6; P(X = $5.00) = 1/4.

where X be a random variable representing the amount of money placed in the money jar by patrons who contribute

a) hence the amount of money the museum gets on average from any individual person who makes a donation is E[X]=1*1/4+2*1/6+3*1/6+4*1/6+5*1/4=$3 [answer]

the shape of the distribution is symmetric since P[X=1]=P[X=5]   and P[X=2]=P[X=4]

hence P[X=E[X]-2]=P[X=E[X]+2] and P[X=E[X]-1]=P[X=E[X]+1]  

so P[X=3-a]=P[X=3+a]   a=1,2..... hence the distribution is symmetric

b) Let’s suppose that 100 people per hour make a donation.

then the average amount of money that the museum will get each hour from donations is 100*E[X]=100*3=$300 [answer]

this is time also the shape would be symmetric

let Y be the random variable denoting the amount donated by 100 people in one hour.

so Y is nothing but Y=100X.

hence Y takes the values 100,200,300,400,500 with probabilities 1/4,1/6,1/6,1/6,1/4 respectively.

withe E[Y]=300

hence P[Y=E[Y]-200]=P[Y=E[Y]+200]   and P[Y=E[Y]-100]=P[Y=E[Y]+100]

hence P[Y=300-a]=P[Y=300+a]   a=100,200

hence the shape is also symmetric here. [answer]

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