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 0. 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: • is uniformly distributed on [0,1]. • O is uniformly distributed on [0,7]. • Z and O are independent and jointly continuous. By geometric reasoning, it can be shown that an intersection occurs if and only if: 1 (z,0) e [0,1] × [0, 7] is such that zssin0 or 1-z< sin0 1 iii) By using the joint distribution function of Z and O,show that: 2 P(The needle intersects a line) = TI 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 < o. 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→ o.
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 0. 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: • is uniformly distributed on [0,1]. • O is uniformly distributed on [0,7]. • Z and O are independent and jointly continuous. By geometric reasoning, it can be shown that an intersection occurs if and only if: 1 (z,0) e [0,1] × [0, 7] is such that zssin0 or 1-z< sin0 1 iii) By using the joint distribution function of Z and O,show that: 2 P(The needle intersects a line) = TI 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 < o. 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→ o.
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 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 [0,7].
• Z and O are independent and jointly continuous.
By geometric reasoning, it can be shown that an intersection occurs if and only if:
1
1
(z,0) € [0,1]×[0, 7] is such that z< sin0 or 1-z< sine
2
iii) By using the joint distribution function of Z and 0, 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? < ∞.
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%2Fe23cd709-0296-4c5a-9214-1d0434132fbb%2F0b091ea3-43fe-49a5-ab88-52b45dd306d0%2Fpnk35q7_processed.png&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 [0,7].
• Z and O are independent and jointly continuous.
By geometric reasoning, it can be shown that an intersection occurs if and only if:
1
1
(z,0) € [0,1]×[0, 7] is such that z< sin0 or 1-z< sine
2
iii) By using the joint distribution function of Z and 0, 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? < ∞.
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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