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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