Suppose that the variblity in a measurement y is modeled by the continous probab

Question

Suppose that the variblity in a measurement y is modeled by the continous probability function pictured below. What is the probability that y exceeds 4? Do not leave your answer as a function of c.

Explanation / Answer

Lets first review some probability basics. If we have a smooth function for probability, the total area under the curve must equal 1, which is because if we sum up all the possible outcomes, we should get 100% of the outcomes (this makes sense intuitively, and so the laws of probability are defined in such a way that makes this true). Knowing this, we know that the graph of the triangle must have an area equal to 1, and so we are left to determine the value indicated as 'c' in the diagram. To do that, we compute the area of a triangle with height 'c' and base 6. Using A = 1/2 (base * height) we know that the total area should be 1, so: 1 = (1/2) * (6) * (c), or 1 = 3 * c, giving us c = 1/3 Now we can find a simple equation that gives us the probability function of the graph. We know that this is a straight line with negative slope, its y-intercept is 1/3 and its x-intercept is 6. Using point-slope formula, we know y - yo = m(x - xo), using the point (0, 1/3) we have y - 1/3 = mx, or y = mx + 1/3 So now we find the slope m using the standard slope formula m = y2 - y1/ x2 - x1. We will use the two points (0, 1/3) and (6, 0) m = (0 - 1/3)/(6 - 0) or m = -1/(3*6) or m = -1/18 So we have our formula! Yay! f(y) = (-1/18)x + 1/3 To finally answer the question, we find out how much area is to the right of y = 4 and that will be the probability that y>4, answering the question. Remember that in this function the 'y' value is on the horizontal axis (very confusing!) So lets find f(4): f(4) = (-1/18) * 4 + 1/3 = 1/3 - 4/18 = 1/3 - 2/9 = 3/9 - 2/9 = 1/9 Next we get the area of the triangle that extends from 4 to 6 on the horizontal axis. It has a base of 2 (6 - 4!) and a height of 1/9 (thats the value of f(4)) A = (1/2) base * height = (1/2) * (1/9) * 2 = 1/9 So the probability that y > 4 is 1/9, or 11.11% Hope that helps! Jonathan

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