Probability , please help

Question

Let X have a uniform distribution on the interval (6, 15). Find the probability that the sum of 2 independent observations of X is greater than 28.

Answer 1

If the sample space is just {6,7,8,9,10,11,12,13,14,15} then the answer is much simpler.

If on the other hand they are REAL NUMBERS, then the solution is much more complicated

with a different answer. If this is the case, skip of this section and start reading BELOW the

row of equal signs.

In this case, the observations are INTEGER numbers between 6 and 15 inclusive. THis is the DISCREET

case, where X = {6,7,8,9,10,11,12,13,14,15} are the ONLY possible outcomes.

There are two possibilities: 1) X=15 and Y=14 or 15

2) X=14 or 15 and Y=15

Prob(X=15) = 1/9. Prob(Y=14 or 15) = 2/9. So this scenario has probability 1/9*2/9 = 2/81

The second possibility has the exact same probability,

THerefore the exact probability is 4/81 which is just under 5%

I modeled this experiment in excel, and created 100 pairs of integers between 6 and 15. Of those

100 pairs, the total number of sums greater than 28 was never more than 5.

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Let x and y be the independent observations, where x,y are uniform.

Since you did not specify that x is an integer, we must assume that x can be any real numbered

value between 6 and 15. That makes the distribution CONTINUOUS, which then requires calculus.

For example, in this case, you can get an outcome of x=7.5 and y = 9.8765

The probability function is f(x) = 1/(15-6) = 1/9 if 6 <=x <=15

0 otherwise

The cumulative distribution function is given by integrating, so F(x) = (1/9)x

The only domain of interest is 6 <=x<=15, so shifting the graph six units to the right gives

Prob(X>x) = F(x) = { (1/9)(x-6) , if 6<= x<15

1 , if x>=15

0 , otherwise

Prob(X=15), using the continuity correction, is F(15) - F(14.5) = 1 - 0.94 = 0.05

Prob(X>13) = (1/9)(7/6) = 7/54 = 0.1296

So the first scenario has probability 0.185

The same probability is for the case Y=15 and X>13. So the total probability thus fr is 0.370

Prob(X>14) and Prob(Y>14) = 2 * [ (1/9)(14-6)/9] = 2 * (1/9)(8/9) = 16/81 = 0.197530864

The total probability is just under 56.8%

THis is because you have a larger range of values that satisfy the condition

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