Joint probability distributions

[jsbeckton]

This is the problem:

Conside a small ferry that can accomidate cars and busses. The toll for cars is $3 and the toll for busses is $10. Let X and Y denote the number of cars and busses respectivly. Compute the expected revenue for the trip.

\(\displaystyle \begin{array}{{}}
{} & {} & {} & y & {} \\
{} & {p(x,y)} & 0 & 1 & 2 \\
{} & 0 & {.025} & {.015} & {.010} \\
{} & 1 & {.050} & {.030} & {.020} \\
x & 2 & {.125} & {.075} & {.050} \\
{} & 3 & {.150} & {.090} & {.060} \\
{} & 4 & {.100} & {.060} & {.040} \\
{} & 5 & {.050} & {.030} & {.020} \\
\end{array}\)

There were a bunch of easy questions asking the probability of certain situations which were easy, but the last one has me stumped (above)


I multiplied all of the x values by their probability sum and add the marginal totals to get x=2.8

Did the same with y and got y=.5

($10)(.5)+($3)(2.8)=$13.40

But the right anwser is $15.40, can anyone tell me where I'm going wrong? Thanks

 

[pka]

 

[jsbeckton]

thanks, could you possibly elaborate on that a bit, is it going to require summing 18 computations?

 

[jsbeckton]

Thanks, I got it, that was a lot of tedeious work!

Another question with the same data being used as the probability distribution of cars and busses at one cycle of a traffic light:

Suppose that a left turn lane can hold 5 cars and a bus counts as 3 cars, what is the probability of an overflow durring a cycle?

would I just add up the probabilities of all situations in which it would overflow?

 

[jsbeckton]

I tried adding all of the probabilities of situations that would lead to an overflow but I got, .38, and it should be .3483?

 

[pka]

I did this with a very inexpensive computer algebra system, StudyWorks (it is a windows program). It is built upon its big brother MathCad. It has most of the functions found in MathCad2000i, except for the D.E. functions and some 3d graphing capabilities. It does not have any of MathCad’s programming capabilities. However, for the cost about $40 (US) it is well worth the money. Images can be copied and pasted into any program.
http://www.learnatglobal.com/html/math_catalog_6.html
Unfortunately, the program is no longer marketed or supported by MathSoft the producer of MathCad!