Download Problem Set 6 File

4CCM141A/5CCM141B – Probability and Statistics I
Exercise Sheet 6
1. The median of a continuous random variable having a Cumulative Distribution Function
(CDF) F ( y ) is the specific value y = m such that F ( m) = 12 . That is, a random variable is
just as likely to be larger than its median as it is to be smaller. Find the median of Y if Y is
a. uniformly distributed over [ a, b] , i.e. Y ~ Uniform [a, b] .
(
)
b. normal with parameters µ ,σ , i.e. Y ~ N µ ,σ 2 .
c. exponential with mean θ , i.e. Y ~ exponential (θ ) .
d. The mode of a continuous random variable having a Probability Density Function (PDF)
p ( y ) is the specific value of y for which p ( y ) attains its maximum.
(i) Compute the mode of Y in the cases a,b,c above.
(ii) What is the value of the density on the mode? Comment on the values you get.
2. A stick of length 1 is split at a point U that is uniformly distributed over ( 0,1) .
a. Determine the expected length of the piece that contains the point p, 0 ≤ p ≤ 1.
b. What is the value of p that maximizes the expected length? What is the corresponding
maximal length?
Hint: you might want to define the following function of the RV U : Lp (U ) = the length of
the substick that contains p . Then find the expectation value of this variable Lp (U ) .
3. (Textbook 4.58+4.59)
a. Use the relevant table to find the following probabilities for a standard normal random
variable Z :
(i) P ( 0 ≤ Z ≤ 1.2 )
(ii) P ( −.9 ≤ Z ≤ 0 )
(iii) P (.3 ≤ Z ≤ 1.56 ) (iv) P ( −.2 ≤ Z ≤ .2)
b. If Z is a standard normal random variable, find the value z0 such that
(i) P ( Z > z0 ) = .5
(ii) P ( Z < z0 ) = .8643
(iii) P ( − z0 < Z < z0 ) = .90
(iv) P ( − z0 < Z < z0 ) = .99
4. (Textbook 4.71) Wires manufactured for use in a computer system are specified to have
resistances between .12 and .14 ohms. The actual measured resistances of the wires
produced by company A have a normal probability distribution with mean .13 ohm and
standard deviation .005 ohm.
a. What is the probability that a randomly selected wire from company A’s production will
meet the specifications?
b. If four of these wires are used in each computer system and all are selected from
company A, what is the probability that all four in a randomly selected system will meet
the specifications?
5. (Textbook 4.75+4.76)
a. A soft-drink machine can be regulated so that it discharges an average of µ ounces per
cup. If the ounces of fill are normally distributed with standard deviation 0.3 ounce, give
the setting for µ so that 8-ounce cups will overflow only 1% of the time.
b. This machine has standard deviation σ that can be fixed at certain levels by carefully
adjusting the machine. What is the largest value of σ that will allow the actual amount
dispensed to fall within 1 ounce of the mean with probability at least .95?
6. (Textbook 4.80) Assume that Y is Normally distributed with mean µ and standard
(
)
deviation σ (i.e. Y ~ N µ ,σ 2 ). After observing a value Y , a mathematician constructs a
rectangle with length L = Y and width W = 3 Y . Let A denote the area of the resulting
rectangle. What is the expected area A ?
7. (Textbook 4.92) The length of time Y necessary to complete a key operation in the
construction of houses has an exponential distribution with mean 10 hours. The formula
C = 100 + 40Y + 3Y 2 relates the cost C of completing this operation to the square of the
time to completion. Find the mean and variance of C.
Chapter 4
Continuous
Variables
and Their Probability Distributions
8. Textbook
4.88, 4.90
The magnitude of earthquakes recorded in a region of North America can be modelled as
an exponential
distributionrandom
with mean
2.4,with
as measured
on the
Richter
scale.
*4.191 having
Suppose
that Y is a continuous
variable
distribution
function
given
by Find
F(y) and
f (y).
We often
interested
theprobability
probabilitydensity
that anfunction
earthquake
striking
this are
region
will in conditional probabilities of the
form P(Y ≤ y|Y ≥ c) for a constant c.
a. exceed 3.0 on the Richter scale.
a Show that, for y ≥ c,
b. fall between 2.0 and 3.0 on the Richter scale.
F(y) − F(c)
P(Y this
≤ y|Y
≥ c)what
= is the probability
.
c. Of the next 10 earthquakes to strike
region,
that at least one
1 − F(c)
will
5.0 on
Richterinscale?
b exceed
Show that
thethe
function
part (a) has all the properties of a distribution function.
c
If the length of life Y for a battery has a Weibull distribution with m = 2 and α = 3 (with
measurements in years), find the probability that the battery will last less than four years,
9. Textbook 4.192 (You can use the fact that Γ(1 / 2) = π )
given that it is now two years old.
*4.192
The velocities of gas particles can be modeled by the Maxwell distribution, whose probability
density function is given by
! m "3/2
2
v 2 e−v (m/[2K T ]) ,
v > 0,
f (v) = 4π
2π K T
where m is the mass of the particle, K is Boltzmann’s constant, and T is the absolute temperature.
a
b
*4.193
Find the mean velocity of these particles.
The kinetic energy of a particle is given by (1/2)mV 2 . Find the mean kinetic energy for a
particle.
Because
F(y) − F(c)
1 − F(c)
Optional
exercises
(from Textbook):
4.18,
4.19+4.25,
4.48,
4.74,
4.91.
has the properties
of a distribution
function,
its derivative
will
have4.51+4.52,
the properties
of4.80,
a probability
density function. This derivative is given by
P(Y ≤ y|Y ≥ c) =
f (y)
,
y ≥ c.
1 − F(c)
We can thus find the expected value of Y , given that Y is greater than c, by using
# ∞
1
E(Y |Y ≥ c) =
y f (y) dy.
1 − F(c) c
If Y , the length of life of an electronic component, has an exponential distribution with mean
100 hours, find the expected value of Y , given that this component already has been in use for
50 hours.
*4.194
We can show that the normal density function integrates to unity by showing that, if u > 0,
# ∞
1
1
2
e−(1/2)uy dy = √ .
√
u
2π −∞
This, in turn, can be shown by considering the product of two such integrals:
$# ∞
% $# ∞
%
# ∞# ∞
1
1
2
2
−(1/2)uy 2
−(1/2)ux 2
e
dy
e
dx =
e−(1/2)u(x +y ) d x d y.
2π
2π −∞ −∞
−∞
−∞
By transforming to polar coordinates, show that the preceding double integral is equal to 1/u.