Let
\[ f_X(x)= \begin{cases} cx, & 0<x<1\\ 0, & \text{otherwise} \end{cases} \](a) Find \(c\).
(b) Compute \(E[X]\).
---Let \(X\) have density \(f_X(x)=2x\) on \(0<x<1\).
(a) Compute \(P(X>0.7)\).
(b) Compute \(E[X]\).
(c) Compute \(\mathrm{Var}(X)\).
---Let
\[ f_{X,Y}(x,y)= \begin{cases} cx, & 0<x<1,\ x<y<1\\ 0, & \text{otherwise} \end{cases} \](a) Find \(c\).
(b) Compute \(f_X(x)\).
(c) Compute \(f_Y(y)\).
(d) Are \(X\) and \(Y\) independent?
---A sensor works as follows:
First, a hidden environmental factor determines the true signal level. This signal level is equally likely to be anywhere between 0 and 2.
Then, the sensor records a measurement by adding random noise to the true signal. The noise can be thought of to have a Normal distribution with mean 0 and variance 1.
Let \(X\) be the recorded measurement.
(a) Compute \(E[X]\).
(b) Compute \(\mathrm{Var}(X)\).
---Let \(X\sim \mathrm{Uniform}(-1,1)\) and \(Y=X^2\).
(a) Compute \(E[X]\), \(E[Y]\), and \(E[XY]\).
(b) Compute \(\mathrm{Cov}(X,Y)\).
(c) Are \(X\) and \(Y\) independent?
---Let \(X\sim \mathrm{Exponential}(\lambda)\).
(a) Compute \(P(X>5 \mid X>2)\).
(b) Compute \(E[X \mid X>2]\).
---Let \(X\sim \mathrm{Uniform}(0,1)\) and define
\[ Y=-\ln(X) \](a) Compute \(P(Y<t)\).
(b) Find the density of \(Y\).
---A factory has two machines.
Before inspecting a batch, you believe each machine is equally likely to have produced it.
A batch is produced entirely by one machine, but the label is missing.
You randomly sample 100 components from the batch and observe that the sample average weight is 11.2.
Based on this information, which machine is more likely to have produced the batch?
What results/tools are you using?
A factory produces items with average weight 10 and variance 4.
You take a sample of 100 items and compute the average weight \(\bar X_n\).
(a) Approximate the probability that \(\bar X_n\) exceeds 10.5.
(b) The inspector chooses a threshold \(c\) so that, only 5% of shipments are rejected. What is this number \(c\).
(c) Suppose that on any given day, there is a 10% chance the machine is faulty.
If the machine is working correctly, the average box weight is 10 grams variance 4.
If the machine is faulty, the average box weight has dropped to 5 grams variance 16.
If the inspector rejects a shipment using the threshold \(c\) from part (b), what is the probability that the machine was actually faulty?