Hi there!
To solve, we can use right triangle trigonometry.
Recall that:
sin = O/H, cos = A/H, tan = O/A.
For angle G, HF is its OPPOSITE side, and FG is the hypotenuse.
Therefore, we must use sine to evaluate:
sinG = 14 / 17
sin⁻¹ (14/17) = ∠G. Evaluate using a calculator.
∠G ≈ 55.44°
A circle with center and radius
has equation
If we substitute in this equation, we have
This equation has two solutions (i.e. the line intersects the circle in two points) if and only if the determinant is greater than zero:
The expression simplifies to
The solutions to the associated equation are
So, the parabola is negative between the two solutions:
I'm going to assume the joint density function is
a. In order for to be a proper probability density function, the integral over its support must be 1.
b. You get the marginal density by integrating the joint density over all possible values of
:
c. We have
d. We have
and by definition of conditional probability,
e. We can find the expectation of using the marginal distribution found earlier.
f. This part is cut off, but if you're supposed to find the expectation of , there are several ways to do so.
The law of total expectation says
We have
This random variable is undefined only when which is outside the support of
, so we have