The prompt is to find the values of missing variables over different types of distribution for the same function.
$$F(t) = \begin{cases}at^{2} &\mbox{if } t\lt 0\\ bt +c &\mbox{0} \leqslant t \lt 2 \\1 &\mbox{t} \geqslant 2 \end{cases}$$
- (a) the distribution function of a random variable,
- (b) the distribution function of a discrete random variable (find the support of the random variable),
- (c) the distribution function of a continuous random variable (find the density of the random variable).
For part a) I began by considering the properties of a random variable being that
The function F(t) is always non-decreasing,
F(x) = $\lim_{x\to -\infty} = 0$ and F(x) = $\lim_{x\to \infty} = 1$,
F(x) is continuous from the right [i.e., $\lim_{h\to 0^+}F(x+h) = F(x)$ for all x]
Unsure on how to use property 1, I begin by property 2, which is proved automatically as $\lim_{t\to -\infty} at^2 = 0$ and $\lim_{t\to \infty} = 1$. Finally I check the continuity of the function at 0 as follows.
$$\lim_{t\to 0^-}at^2 = \lim_{h\to 0}a(0-h)^2 = 0$$ $$\lim_{t\to 0^+}bt+c = \lim_{h\to 0}b(0+h) + c = c$$ Since function is continuous, c = 0.
Similarly, finding the limit at 2, $$\lim_{t\to 2^-}bt + c = \lim_{h\to 0}b(2-h) + c = 2b + c$$ $$\lim_{t\to 2^+} = 1$$ This gives us 2b + c = 1, since c = 0, $b = \frac{1}{2}$ I'm not sure on how to proceed finding value of a or follow with other parts of the question since they are interdependent.