The amplitude of a certain audio signal is modeled by a continuous random variable X X\,X with a probability density function f(x)f(x)f(x) defined as:
f(x)={19(x+1)−1<x≤2162<x≤50otherwise f(x) = \begin{cases} \frac{1}{9}(x+1) & -1 < x \le 2 \\ \frac{1}{6} & 2 < x \le 5 \\ 0 & \text{otherwise} \end{cases} f(x)=⎩⎨⎧91(x+1)610−1<x≤22<x≤5otherwiseSketch the graph of f(x)f(x)f(x) for all xxx.
Determine the conditional probability P(1.5≤X≤4∣X≥0)P(1.5 \le X \le 4 \mid X \ge 0)P(1.5≤X≤4∣X≥0).
The random variable Y Y\,Y represents the measurement noise power in the same circuit, where E(Y)=4E(Y) = 4E(Y)=4 and Var(Y)=2Var(Y) = 2Var(Y)=2.
Find E(4Y2−8X+3)E(4Y^2 - 8X + 3)E(4Y2−8X+3).
Show your working clearly. Solutions relying solely on calculator technology for integration or statistical distributions are not acceptable.
