mly “drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre en by two random co-ordinates (X,Y), and the angle is given (in radians) by a om variable O. In this question, we will be concerned with the probability that eedle intersects one of the lines y = n. For this purpose, we define the random ple Z as the distance from the needle's centre to the nearest line beneath it (i.e. - [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, 1] . Z and O are independent and jointly continuous. Give the density functions of Z and O. Give the joint density function of Z and O (hint: use the fact that Z and O are ndependent). ometric reasoning, it can be shown that an intersection occurs if and only if:
mly “drop a needle" (i.e. draw a line segment) of length 1 on the plane: its centre en by two random co-ordinates (X,Y), and the angle is given (in radians) by a om variable O. In this question, we will be concerned with the probability that eedle intersects one of the lines y = n. For this purpose, we define the random ple Z as the distance from the needle's centre to the nearest line beneath it (i.e. - [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, 1] . Z and O are independent and jointly continuous. Give the density functions of Z and O. Give the joint density function of Z and O (hint: use the fact that Z and O are ndependent). ometric reasoning, it can be shown that an intersection occurs if and only if:
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
![Consider a two-dimension plane in which we mark the lines y = n for ne 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, 77].
• 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:
1
(z, 0) € [0,1]x [0, 7] is such that z< sin 0 or 1-z< sin 0
1
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 o? < o.
iv) Explain, with reference to the Law of Large Numbers, how the statistician could
use this experiment to estimate the value of T with increasing accuracy.
v) Explain what happens to the distribution of X as n →o.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ab89ad2-328d-4004-b31e-cca641a10119%2F239f1220-8da3-4c27-a7d3-9de7be198f74%2F1sjwd16_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider a two-dimension plane in which we mark the lines y = n for ne 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, 77].
• 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:
1
(z, 0) € [0,1]x [0, 7] is such that z< sin 0 or 1-z< sin 0
1
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 o? < o.
iv) Explain, with reference to the Law of Large Numbers, how the statistician could
use this experiment to estimate the value of T with increasing accuracy.
v) Explain what happens to the distribution of X as n →o.
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