Home Schedule **Homework** | | | | # Homework ## Homework 6, due Dec 9 at 2:07pm Please do the following problems from the textbook: DeGroot-Schervish 3.9, p187: 7, 14, 19 DeGroot-Schervish 4.1, p216: 7 DeGroot-Schervish 4.2, p224: 4, 5, 11 DeGroot-Schervish 4.3, p233: 5 DeGroot-Schervish 4.9, p272: 4 And the following additional problems: **A.** Let *X* and *Y* be independent continuous random variables with pdfs, and let *phi* and *psi* be strictly increasing, differentiable functions. Find the joint pdf of *phi*(*X*) and *psi*(*Y*). Conclude that *phi*(*X*) and *psi*(*Y*) are independent. **B.** If *X* is an exponential random variable with parameter *c* (recall this means pdf *ce*^{-cx} for *x* > 0, zero elsewhere), then find the expected value and variance of *X*. Hint for p187 #7: first find the pdf of -*X*_{2}. Hint for p224, #11: Example 4.2.9 is relevant to finding the expected value of *X*_{i}. ## Homework 5, due Nov 30 at 2:07pm Please do the following problems from the textbook: ~~DeGroot-Schervish 3.4, p129: 10~~ DeGroot-Schervish 3.5, p141: 14, 15 DeGroot-Schervish 3.6, p151: 6 DeGroot-Schervish 3.7, p167: 4, 5 DeGroot-Schervish 3.8, p174: 2, 8, 11 DeGroot-Schervish 3.9, p187: 1, 6 And the following additional problems: **A.** You choose a point uniformly at random from the interior of the unit circle (the unit circle dartboard example from class). Let *R* be the radius and *T* the angle in a polar coordinates expression for this point. Find the joint cdf of *R* and *T*. Are *R* and *T* independent? **B.** If *X* is a random variable with cdf *F*, and *phi* is a strictly decreasing function on the support of *F*, then find the cdf of *phi*(*X*). If, in addition, *X* has a pdf and *phi* is differentiable, then find the pdf of *phi*(*X*). **C.** If *X* is a (continuous) uniform random variable on [-1,2], then find the cdf and pdf of *X*^{2}. Hint: It may be helpful to condition on the sign of *X*. ## Homework 4, due Nov 14 at 2:07pm Please do the following problems from the textbook: DeGroot-Schervish 3.1, p100: 9, 11 DeGroot-Schervish 3.2, p107: 6, 8, 10 DeGroot-Schervish 3.3, p116: 4, 6, 12, 17ab DeGroot-Schervish 3.4, p129: 4, 8, 10 |