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.
200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.