[SOLVED] Probability & statistics for eecs homework 07

1. Let
F(x) = 2
π
sin−1
(
√x), for 0 < x < 1,
and let F(x) = 0 for x ≤ 0 and F(x) = 1 for x ≥ 1.
(a) Check that F is a valid CDF, and find the corresponding PDF f.(b) Explain how it is possible for f to be a valid PDF even though f(x) goes to ∞ as
x approaches 0 and as x approaches 1.2. Let F be a CDF which is continuous and strictly increasing. Let µ be the mean of
the distribution. The quantile function, F −1, has many applications in statistics and
econometrics. Show that the area under the curve of the quantile function from 0 to 1
is µ.3. Let U1, . . . , Un be i.i.d. Unif(0, 1), and X = max(U1, …, Un). What is the PDF of X?
What is E(X)?4. A stick of length 1 is broken at a uniformly random point, yielding two pieces. Let X
and Y be the lengths of the shorter and longer pieces, respectively, and let R = X/Y
be the ratio of the lengths X and Y .(a) Find the CDF and PDF of R.
(b) Find the expected value of R (if it exists).
(c) Find the expected value of 1/R (if it exists).5. The Exponential is the analog of the Geometric in continuous time. This problem
explores the connection between Exponential and Geometric in more detail, asking
what happens to a Geometric in a limit where the Bernoulli trials are performed faster
and faster but with smaller and smaller success probabilities.Suppose that Bernoulli trials are being performed in continuous time; rather than only
thinking about first trial, second trial, etc., imagine that the trials take place at points
on a timeline.Assume that the trials are at regularly spaced times 0, ∆t, 2∆t, . . . ,
where ∆t is a small positive number. Let the probability of success of each trial be
λ∆t, where λ is a positive constant. Let G be the number of failures before the first
success (in discrete time), and T be the time of the first success (in continuous time).(a) Find a simple equation relating G to T.
(b) Find the CDF of T.
(c) Show that as ∆t → 0, the CDF of T converges to the Expo(λ) CDF, evaluating
all the CDFs at a fixed t ≥ 0.6. Let Z ∼ N (0, 1), and c be a nonnegative constant. Find E(max(Z − c, 0)), in terms
of the standard Normal CDF Φ and PDF ϕ.