Let $X$ be a non-negative random variable such that $\mathbb{E}\left[X^{2}\right]>0$ is finite, and let $\theta \in[0,1]$.

(a) Show that

$$\mathbb{E}[X \mathbb{I}[\{X>\theta \mathbb{E}[X]\}]] \geqslant(1-\theta) \mathbb{E}[X]$$

(b) Let $Y_{1}$ and $Y_{2}$ be random variables such that $\mathbb{E}\left[Y_{1}^{2}\right]$ and $\mathbb{E}\left[Y_{2}^{2}\right]$ are finite. State and prove the Cauchy-Schwarz inequality for these two variables.

(c) Show that

$$\mathbb{P}(X>\theta \mathbb{E}[X]) \geqslant(1-\theta)^{2} \frac{\mathbb{E}[X]^{2}}{\mathbb{E}\left[X^{2}\right]}$$