 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/14/c/i/solution

Y = y

X

equal to

f

. Given that value of

Y

Z

is the translate

X + y

. Hence

P (z \leq Z \leq z + d z ∣ Y = y) = \int_{z - y}^{z + d z - y} f (x) d x .

At a continuity point of

f

, the expression is

f (z - y) d z + o (d z)

. Since

P (Y = y) = 0

Y = y

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