Consider a two-dimension plane in which we mark the lines y = n for nɛ Z. We now randomly "drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre is given by two random co-ordinates (X,Y), and the angle is given (in radians) by a random variable O. In this question, we will be concerned with the probability that the needle intersects one of the lines y = n . For this purpose, we define the random variable Z as the distance from the needle's centre to the nearest line beneath it (i.e. Z = Y – [Y], where [Y] is the greatest integer not greater than Y ). We assume: • Z is uniformly distributed on [0,1]. O is uniformly distributed on [0,7]. • Z and O are independent and jointly continuous. i) Give the density functions of Z and O. ii) Give the joint density function of Z and O (hint: use the fact that Z and O are independent).
Consider a two-dimension plane in which we mark the lines y = n for nɛ Z. We now randomly "drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre is given by two random co-ordinates (X,Y), and the angle is given (in radians) by a random variable O. In this question, we will be concerned with the probability that the needle intersects one of the lines y = n . For this purpose, we define the random variable Z as the distance from the needle's centre to the nearest line beneath it (i.e. Z = Y – [Y], where [Y] is the greatest integer not greater than Y ). We assume: • Z is uniformly distributed on [0,1]. O is uniformly distributed on [0,7]. • Z and O are independent and jointly continuous. i) Give the density functions of Z and O. ii) Give the joint density function of Z and O (hint: use the fact that Z and O are independent).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Consider a two-dimension plane in which we mark the lines y = n for nɛ Z. We now
randomly "drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre
is given by two random co-ordinates (X,Y), and the angle is given (in radians) by a
random variable O. In this question, we will be concerned with the probability that
the needle intersects one of the lines y = n . For this purpose, we define the random
variable Z as the distance from the needle's centre to the nearest line beneath it (i.e.
Z = Y – [Y], where [Y] is the greatest integer not greater than Y ). We assume:
Z is uniformly distributed on [0,1].
O is uniformly distributed on [0,7].
• Z and O are independent and jointly continuous.
i) Give the density functions of Z and O.
ii) Give the joint density function of Z and O (hint: use the fact that Z and O are
independent).
By geometric reasoning, it can be shown that an intersection occurs if and only if:
(z, 0) E [0,1]× [0, 1] is such that z<
1
sin 0 or 1 –z<
1
sin 0
iii) By using the joint distribution function of Z and O, show that:
2
P(The needle intersects a line)
Suppose now that a statistician is able to perform this experiment n times without
any bias. Each drop of the needle is described by a random variable X; which is 1
if the needle intersects a line and 0 otherwise. For any n, we assume the random
variables X1,..., X, are independent and identically distributed and that the variance
of the population is o² < ∞.
iv) Explain, with reference to the Law of Large Numbers, how the statistician could
use this experiment to estimate the value of n with increasing accuracy.
v) Explain what happens to the distribution of X as n →∞.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ab89ad2-328d-4004-b31e-cca641a10119%2Fb8c62935-057c-4f80-b15f-83f9ec52ec9e%2F9ek6n2s_processed.png&w=3840&q=75)
Transcribed Image Text:Consider a two-dimension plane in which we mark the lines y = n for nɛ Z. We now
randomly "drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre
is given by two random co-ordinates (X,Y), and the angle is given (in radians) by a
random variable O. In this question, we will be concerned with the probability that
the needle intersects one of the lines y = n . For this purpose, we define the random
variable Z as the distance from the needle's centre to the nearest line beneath it (i.e.
Z = Y – [Y], where [Y] is the greatest integer not greater than Y ). We assume:
Z is uniformly distributed on [0,1].
O is uniformly distributed on [0,7].
• Z and O are independent and jointly continuous.
i) Give the density functions of Z and O.
ii) Give the joint density function of Z and O (hint: use the fact that Z and O are
independent).
By geometric reasoning, it can be shown that an intersection occurs if and only if:
(z, 0) E [0,1]× [0, 1] is such that z<
1
sin 0 or 1 –z<
1
sin 0
iii) By using the joint distribution function of Z and O, show that:
2
P(The needle intersects a line)
Suppose now that a statistician is able to perform this experiment n times without
any bias. Each drop of the needle is described by a random variable X; which is 1
if the needle intersects a line and 0 otherwise. For any n, we assume the random
variables X1,..., X, are independent and identically distributed and that the variance
of the population is o² < ∞.
iv) Explain, with reference to the Law of Large Numbers, how the statistician could
use this experiment to estimate the value of n with increasing accuracy.
v) Explain what happens to the distribution of X as n →∞.
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