- Please Answer Iii Iv V 1 (106.27 KiB) Viewed 107 times
please answer (iii) (iv) (v)
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please answer (iii) (iv) (v)
please answer (iii) (iv) (v)
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 e. 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]. • is uniformly distributed on (0,7). • Z and are independent and jointly continuous. i) Give the density functions of Z and e. ii) Give the joint density function of Z and (hint: use the fact that Z and are independent). By geometric reasoning, it can be shown that an intersection occurs if and only if: s sino 1 (2,0) € [0, 1] x [0,1] is such that zs-sino or 1-25-sino 2 iii) By using the joint distribution function of Z and show that: 2 P(The needle intersects a line) = 70 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 <0. 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 11 →
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 e. 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]. • is uniformly distributed on (0,7). • Z and are independent and jointly continuous. i) Give the density functions of Z and e. ii) Give the joint density function of Z and (hint: use the fact that Z and are independent). By geometric reasoning, it can be shown that an intersection occurs if and only if: s sino 1 (2,0) € [0, 1] x [0,1] is such that zs-sino or 1-25-sino 2 iii) By using the joint distribution function of Z and show that: 2 P(The needle intersects a line) = 70 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 <0. 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 11 →