Download Bowdoin Math 2606, Spring 2016 – Homework #3

Bowdoin Math 2606, Spring 2016 – Homework #3
Joint c.d.f., p.f./p.d.f., median, independence, the Normal and χ2 p.d.f.’s.
Due on Wednesday, February 17 (at the beginning of class)
This homework assignment is on two pages.
Read sections1 3.5 and 5.6 of Probability and Statistics by DeGroot and Schervish. Also, solve the following
problems, some of which may be taken from the textbook. To obtain full credit, please write clearly and
show your reasoning. Please solve the problems in the order given and STAPLE your homework
together. Also, write your name (LAST, First) on the top of the first page.
Problem 3A (joint c.d.f.’s). Suppose that X and Y are random variables such that (X, Y ) must belong
to the rectangle in the xy-plane containing all points (x, y) for which 0 ≤ x ≤ 3 and 0 ≤ y ≤ 4. Suppose
also that the joint c.d.f. of X and Y at every point (x, y) in this rectangle is specified as follows:
F (x, y) =
1
xy(x2 + y).
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Determine:
(a) P (1 ≤ X ≤ 2 and 1 ≤ Y ≤ 2),
(b) P (2 ≤ X ≤ 4 and 2 ≤ Y ≤ 4),
(d) the joint p.d.f. of X and Y ,
(e) P (X ≤ Y ).
(c) the c.d.f. of Y,
Problem 3B (joint p.f./p.d.f.’s). Let Y be the rate (calls per hour) at which calls arrive at a switchboard.
Let X be the number of calls during a two-hour period. A popular choice of joint p.f./p.d.f. for (X, Y ) in
this example would be one like:
(2y)x
−3y
if y > 0 and x = 0, 1, 2, 3, . . . ,
x! e
fX,Y (x, y) =
0
otherwise
(a) Verify that f is a joint p.f./p.d.f. (Hint: sum over the x values first; you will need a Taylor series.)
(b) Find P (X = 0).
Problem 3C (mean vs. median). The median m of a continuous random variable X is its 12 -quantile,
i.e. the number m such that P (X ≤ m) = 12 . Compute the median of a random variable X ∼ Exp(λ). Is it
smaller or greater than its mean E(X)? Can you give an intuitive explanation of why this is the case, based
on the shape of the graph of the p.d.f.?
Problem 3D (mean and variance of Normal random variables). Consider first a standard Normal
1 2
random variable Z ∼ N (0, 1), i.e. with p.d.f. ϕ(z) = √12π e− 2 z , with z ∈ R.
R∞
(a) Show that, as claimed in class, E(Z) = 0 and Var(Z) = 1. Hint: to compute E(Z 2 ) = −∞ z 2 ϕ(z) dz
one option to use integration by parts, with u = z and v 0 = zϕ(z).
(b) Now, for fixed constants µ ∈ R and σ > 0, define X = σZ + µ. First let’s show that X ∼ N (µ, σ 2 ). To
x−µ
do so first compute the c.d.f. of X, i.e. FX (x) = P (X ≤ x) = P (σZ + µ ≤ x) = P (Z ≤ x−µ
σ ) = Φ( σ );
note that the inequality “≤” stays the same because we have divided by σ, which is a positive number.
Then compute fX = FX0 , applying the chain rule.2
(c) Finally, use part (a) and the linearity of the expectation to show that E(X) = µ and that Var(X) = σ 2 .
1
Note that in section 5.6 the textbook uses the moment generating function, which is a tool that we have not developed yet,
to compute mean and variance of a Normal random variable. You will compute them in the ‘classical’ way in this assignment.
2
d
Remember that the chain rule gives the derivative of “nested” functions: dx
h(`(x)) = h0 (`(x)) `0 (x).
1
Problem 3E. Consider a random variable Z ∼ N (0, 1), i.e. with a standard normal probability density
function. Compute the following probabilities, using the tabulated standard normal cumulative distribution
function Φ(z) tabulated on page 861 of Probability and Statistics by DeGroot and Schervish (alternatively,
you may use the one that was handed out in class):
(a) P (Z < 1.48)
(b) P (Z < −1.35)
(c) P (−0.4 < Z < 1.5)
(d) P (Z > −1.13)
(e) P (|Z| > 2)
Problem 3F (LA weather). The average daily high temperature in Los Angeles is µ = 77◦ F with a
standard deviation of σ = 5◦ F. Suppose that the temperatures in June closely follow a normal distribution;
that is, if we let X be the temperature of a randomly chosen day in June, we have that X ∼ N (µ, σ 2 ).
(a) What is the probability of observing an 83◦ F temperature or higher during a randomly chosen day in June?
(b) What is instead the probability of observing a temperature between 70◦ F and 80◦ F?
(c) How cold are the coldest 10% of the days during June in LA?
(d) Let Y be temperature of a randomly chosen day in June measured in Celsius degrees (◦ C) instead of
Fahrenheit degrees: that is, Y = (X − 32) × 95 . Write the probability model for Y .
(e) What is the probability of observing a 28.33◦ C (which corresponds to 83◦ F) temperature or higher in
June in LA? Calculate it using the ◦ C model (i.e. the one for the random variable Y ) from part (d).
Did you get the same answer as in part (a) or not? Are you surprised? Explain.
Problem 3G (independence). In class we showed that if two continuous random variables X and Y are
independent (that is, if P (X ∈ A, Y ∈ B) = P (X ∈ A) · P (X ∈ B) for any sets A ⊆ R and B ⊆ R) then
their joint p.d.f. is equal to the product of the marginal p.d.f.’s, i.e. fX,Y (x, y) = fX (x)fY (y), for (x, y) ∈ R2 .
(a) Show that the opposite holds: if fX,Y (x, y) = fX (x)fY (y), for all (x, y) ∈ R2 , then X, Y are independent.
(b) Now show that if X and Y are independent, then for any pair of functions g : R → R and h : R → R we
have that E[g(X)·h(Y )] = E[g(X)]·E[h(Y )] (so, for example, we have that E[X n Y m ] = E[X n ]E[Y m ],
for any values of n and m). Hint: use the factorization of the joint p.d.f., fX,Y (x, y) = fX (x)fY (y).
Problem 3H (expectation and p.d.f. of a function of a random variable, Y = g(X)). Consider a
random variable X ∼ Uniform[e, e2 ] and define the new random variable Y = ln X.
R∞
(a) Compute E(Y ), using the theorem that states that E(g(X)) = −∞ g(x)fX (x) dx.
(b) Now we will calculate E(Y ) by computing the probability density function of Y first. To do so the
initial step is to figure out what the c.d.f. of Y is, by noticing that:
(∗)
FY (y) = P (Y ≤ y) = P (ln X ≤ y) = P (X ≤ ey ) = FX (ey ).
(1)
Note that in step (∗) we have applied the exponential function to both sides of the inequality, and
used the fact that such function is monotone increasing (that is, a ≤ b if and only if ea ≤ eb . Had we
used a monotone decreasing function, we would have had to switch the direction of the inequality!).
3
To find the p.d.f. of Y apply the chain
R ∞ rule to (1). Now that you have fY , compute the expectation
of Y via the usual formula E(Y ) = −∞ yfY (y) dy. You should get the same result as in part (a).
(c) Now let’s spice things up a little and use a function g that is not monotone increasing or decreasing:
consider a standard normal random variable Z ∼ N (0, 1), and T = g(Z) = Z 2 . First note that T ≥ 0,
so its p.d.f. fT (t) will be zero for t < 0. To compute it, we first find the c.d.f.; for t ≥ 0 we will have:
√
√
√
√
FT (t) = P (T ≤ t) = P (Z 2 ≤ t) = P (− t ≤ Z ≤ t) = Φ( t) − Φ(− t).
Finally, use the chain rule to compute the p.d.f. fT , and roughly sketch its graph. This is the so-called χ2
(“chi-squared”) probability density function, which is of fundamental importance in statistics.
3
See the previous footnote.
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