Help with statistics problem (1 Viewer)

Trebla

Let D be the detected speed in the camera, S be the actual speed and E be the error such that E = D - S.

We are looking for the event that an infringement occurs (i.e. D > 80) and the actual speed is does not exceed 80 km/h (S < 80) for the two different values of the errors.

If the detected speed D is 2 km/h above the actual speed, then this event occurs when the actual speed S is between 78 km/h and 80 km/h.
If the detected speed D is 3 km/h above the actual speed, then this event occurs when the actual speed S is between 77 km/h and 80 km/h.
If the detected speed D is 1 km/h below the actual speed, then no infringement occurs

Hence, for part (i), we are looking for the following probability

$\bg_white P((78 < S \leq 80) \cap E = 2) + P((77 < S \leq 80) \cap E = 3)$
$\bg_white = P(78 < S \leq 80)P(E = 2) + P(77 < S \leq 80)P(E = 3)$

Noting that the actual speed of the vehicle is independent of the type of error in the speed camera. For the second part, we are only looking at the first term of E = 2, as the event of E = -1 does not lead to an infringement (given S < 80) .

For part (ii), we are looking for the event of not having an infringement (i.e. D < 80) and the actual speed exceeds 80 km/h (S > 80)

If the detected speed D is 2 km/h above the actual speed, then an infringement will always occur for S > 80.
If the detected speed D is |2+α| km/h below the actual speed, then no infringement occurs provided 80 < S < 80-(2+α)

Note that a necessary condition for this to occur is α < -2 as the detected speed must always be below the actual speed to have no infringement if the actual speed already exceeds the limit of 80 km/h.

Hence, we are looking for the following probability

$\bg_white P((80 < S < 78 - \alpha) \cap E = 2+\alpha)$
$\bg_white = P(80 < S < 78 - \alpha)P(E = 2+\alpha)$

Let this equal to 7.125% and solve to get the boundary value for α (which should be negative). The range of values of α will then be between -2 and this value.