Exercises Distributions P&S I
1.
2.
Discrete Distributions
We throw a dice twice (an unbiased dice, independent throws)
a. Give the sample space of this experiment and the probability of every outcome.
b. If X = “the sum of both results” , determine the outcomes in the event {X = 7}.
c. Determine the distribution of X (the probability function of X).
d. Compute E(X) and σX.
The life expectancy of a machine (in completed years) can be modelled with the following
distribution:
x
P(X= x)
0
0.05
1
0.25
2
0.30
3
0.20
4
0.10
5
0.05
6
0.05
a.
3.
4.
5.
6.
7.
Compute the probability that the machine will not break down during at least 5 years under the
condition that it has survived 3 years.
b.
Compute the expected value of the life expectancy .
c.
Compute the standard deviation of the life expectancy.
X has a homogeneous distribution on {1, 2, 3, ..., N}: each outcome is equally likely. Compute E(X)
Which of the following problems can be modelled with the binomial distribution? If so, which
parameters?
a.
X is the number of defective machines if we buy 4 arbitrary machines of a manufacturer who has
three defective and seven good machines in stock.
b.
Compute the probability that at least 9 out of 40 persons respond to a mailing if usually about 15%
respond.
c.
A machine is subject to wear so that at the end of the day it produces more failures. We want to
compute the probability of at least one failure if we choose 10 products from a day`s production.
d.
Data from the past prove that 10% of the clients of car insurance company claim money in a year. A
random sample of 200 clients is sent a questionnaire. X is the number of these clients who have
submitted a claim last year.
X has a B(25, 0.2)-distribution.
Calculate a. E(X2)
b. P(X > 2)
Observe Chebyshevs inequality:
a.
If X has a sample space SX = {−1, 0, 1}, determine a distribution that maximizes the variance of X.
b.
Use the distribution of a. to check for which c > 0 the equality in Chebyshevs inequality holds.
c.
If Z is geometrically distributed with parameter p = 9/25, check Chebyshevs inequality for c = 2σZ.
Show that the variance of the geometrical distribution is
. Hint: first use the second derivative of the
geometrical series to compute E[X(X+1)] and then use the identity var(X) = E[X(X+1)] – EX − E(X) 2.
A company has 150 employees and wants to install telephone exchange equipment with n lines
(connections) outside the company’s building. The higher n, the higher the costs. Assume that at arbitrary
moments (on the average) one of fifty employees is calling outside the company.
a. What distribution(s) would you recommend to model the distribution of X ,the number of employees
who want to use an outside line at an arbitrary moment (name the “exact” and the approximating
distribution) ?
b. Determine the smallest number of lines (outside the company), so that the probability of overload of
the lines at an arbitrary moment is smaller than 5%. (Use a Poisson table if available)
9. (Key problem) John is trying to open a locked door using a bunch of N = 10 keys. Determine the
probability function of X = “the number of trials, necessary to open the door” and compute E(X) if :
a. John chooses an arbitrary key from the bunch of 10 keys at every trial.
b. John chooses at every trial a key from the remaining keys (without replacement)
10. Assign to each of the following situations an appropriate probability model (state the distribution of X)
and compute the probability P(X > EX).
a. A travelling salesman sells products to (on the average) 15% of the clients he is visiting. On a day he
has a list of 12 (potential) clients: X is the number of clients he sells products to.
b. A hospital has two incubators (intensive care units for babies) available. The mean daily demand of
incubators is two as well. X is the demand of incubators on an arbitrary day.
c. A group of 10 astronauts, consisting of 3 women and 7 men, is training for a voyage in space. Four of
them are selected at random for next month’s space trip. X is the number of chosen women.
11. From previous research it is known that about 4% of the eggs of weight class 1(70 grams and larger) in a
supermarket has a lower weight. These eggs are sold in boxes of 10 eggs. Compute or approximate:
8.
a.
The probability that at least one egg in a box is not a weight-class-1-egg (What are your assumptions
for this computation?) .
b. The probability that ten boxes of 10 eggs together contain 4 or more eggs outside weight class 1.
c. The expected number of boxes, if we buy boxes of eggs one by one as long as all eggs are weightclass-1-eggs
12. a. In a game we have to throw 3 sixes with one dice (not necessarily 3 times 6 in a row)
What is the probability that we throw the third 6 in the tenth trial?
b. (Generalization of a.) If we repeatedly execute Bernoulli trials (success probability p) what is the
probability of the k-th success in the n-th trial?
13. The probability function of X is
, k = 0, 1 , 2 ,…. (c > 0 is constant)
a.
b.
Compute the value of c.
What transformation could you use to transform this probability function to one of the common
discrete distributions?
c. Compute E(X).
14. The number M is called the median of a distribution if P(X ≥ M) ≥ ½ and P(X ≤ M) ≥ ½
The number m is called the mode (modus) if P(X = m) is the largest probability
Find the median and mode for
a. The geometrical distribution having parameter p = 1/3
b. The Poisson distribution having parameter µ = 2.5
Continuous Distributions
15. X has an exponential distribution with unknown parameter λ.
a. Show that
and sketch the graph of f.
16.
17.
18.
19.
20.
21.
22.
b. Show that E(X) = 1/λ and that E(Xn) = n!/λn
c. Compute P(X > EX). Does the result depend on λ?
d. Compute the median M (the value for which P(X > M) = 0.50 and compare to the expectation.
e. Determine the mode (the value of X for which f(x) is largest)/
X has a uniform distribution on [a, b]
a. Derive the formula of var(X).
b. Find the (cumulative) distribution function of X.
X ~ N(0, 1), Y ~ N(1, 4) and Z ~ N(μ, σ2). Use the standard normal table to compute:
a. P(X ≤ 3), P(X ≤ −2), P(−2 < X < 3) and P(−2 ≤ X ≤ 3).
b. P(Y ≤ 7) and P(Y ≤ −3).
c. P(μ − σ ≤ Z ≤ μ + σ)
d. P((Y − 3) 2 ≤ 16)
The length (in cm) of an arbitrary Dutch man can be considered to be a N(182, 50)-distributed variable.
What is the percentage of Dutch men having a length between 175 and 185 cm.
The diameter of green peas can be modelled as a r.v. X, having a N(6, 2.25)-distribution. Using sieves the
producer of peas wants to sell the peas in 4 classes: “large” , “middle” , “fine” and “very fine”. These
classes should be 10, 40, 30 and 20% of the total number of peas respectively. What size should the sieves
have to establish these proportions? (Hint: sketch the normal distribution and the desired relative
frequencies of the classes as area`s)
The random variable X has a distribution function F(x) = 1- ae-bx for x ≥ 0 (
) and F(x) = 0 if x < 0.
a. For what values of a and b is F(x) a distribution function?
b. For what value(s) of a is X a continuous random variable?
c. If b =1, for what value(s) of a is X a discrete r.v.?
X has a density function defined by: f(x) = cxe−x if x ≥ 0 (c
)
and f(x) = 0 if x < 0
a. Determine the value of c and sketch the graph of f.
b. Determine the distribution function of X.
c. Compute E(X)
a. Proof the identity E(aX + b) = a E(X) + b for a continuous random variable.
b. Proof
, if Y = aX + b and a < 0.
23. X ~ U[0, c] (uniform on the interval [0, c] , c > 0)
a. Find an expression for the k-th moment E(Xk) and use it to compute σX.
b. Compute E(X - µX)3.
E(X - µX)3 is a measure for skewness, so in the case of a symmetrical density function it should be 0.
24. a. X ~ N(0, 1). Derive
if Y = eX. (the distribution of Y is called log-normal, applied in economy).
b. X ~ U(0, 1). Derive
if Y = - ln (X) / λ. (Used to generate exponentially distributed times)
c. X ~ U(-1, 1). Derive
if Y = X 2 .
(Note that Y has the same sample space as X but not the same distribution)
25. X is a random number drawn from the interval (0, 1) and Y is its reciprocal:
a.
b.
Find the density function of Y.
Compute, if possible, in two ways (using the density of X resp. Y):
1. P(Y > 2 )
and
2. E(X).