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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).

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).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.2: Trigonometric Equations
Problem 97E
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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 →∞.
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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