Let $X$ and $Y$ be two real-valued random variables with

$$ E(X)=1, E(Y)=2, E\left(X^2\right)=4, E\left(Y^2\right)=9, \text { and } E(X Y)=0.9, $$

where $E$ denotes the expectation operator.

The value of $\alpha$ that minimizes $\mathrm{E}\left((\mathrm{X}-\alpha \mathrm{Y})^2\right)$ is $\_\_\_\_$ .

(Round off to one decimal place)

Let $X$ and $Y$ be continuous random variables with probability density functions $P_X(x)$ and $P_Y(y)$, respectively. Further, let $Y=X^2$ and $P_X(x)=\left\{\begin{array}{cc}1, & x \in(0,1] \\ 0, & \text { otherwise }\end{array}\right.$.

The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is __________ (Round off to 3 decimal places).

Let the probability density function of a random variable x be given as

f(x) = ae^$$-$$2|x|

The value of a is _________.