An archer shoots an arrow at a circular target of radius 50 cm. The arrow pierces the target at a random position (X,Y), measured in centimeters from the center of the disk at position (X,Y) = (0,0). The bullseye is a solid black circle of radius 2 cm, at the center of the target. Calculate the probability P[B] of the event that the archer hits the bullseye under each of the following models: a) X and Y are iid continuous uniform (-50,50) random variables. b) The PDF fX,Y(x,y) is uniform over the 50 cm circular target. c) X and Y are iid Gaussian (μ=0, σ=10) random variables.

An archer shoots an arrow at a circular target of radius 50 cm. The arrow pierces the target at a random position (X,Y), measured in centimeters from the center of the disk at position (X,Y) = (0,0). The bullseye is a solid black circle of radius 2 cm, at the center of the target. Calculate the probability P[B] of the event that the archer hits the bullseye under each of the following models: a) X and Y are iid continuous uniform (-50,50) random variables. b) The PDF fX,Y(x,y) is uniform over the 50 cm circular target. c) X and Y are iid Gaussian (μ=0, σ=10) random variables.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.7: More On Inequalities

Problem 44E

See similar textbooks

Similar questions

A straight line is drawn at random through a point (0, 1) (Figure 22). Find the probability density of the random variable n = 1 cos p. A A(0,2) FIGURE 22
When studying the spread of an epidemic, we assume that the probability that an infected individual will spread the disease to an uninfected individual is a function of the distance between them. Consider a circular city of radius 41 mi in which the population is uniformly distributed. For an uninfected individual at a fixed point A(x0, y0), assume that the probability function is given, d(P, A) denotes the distance between P and A. f(P) = (1/82) * 82 − d(P, A) (a) Suppose the exposure of a person to the disease is the sum of the probabilities of catching the disease from all members of the population. Assume that the infected people are uniformly distributed throughout the city, with k infected individuals per square mile. Find a double integral that represents the exposure of a person residing at A(x0, y0).(b) Evaluate the integral for the case in which A is the center of the city and for the case in which A is located on the edge of the city. Where is the risk of infection lower?…
b) Let Z₁-N(0,1), and W₁ = Y~N(0,1), for i=1,2,3,...,10, then: dx dy i) State, with parameter(s), the probability distribution of the statistic, T = - 154 ii) Find the mean and variance of the statistic T = ₁² 10 iii) Calculate the probability that a statistic T = Z₁ + W₁ is at most 4. iv) Find the value of ß such that P(T> B) = 0.01, where T = ₁2₁² +².
The length of time to failure (in hundreds of hours) for a transistor is a random variable Y with cumulative distribution function given by J1-e, y20 F, (y) = |0, у100|Y < 200).
We are throwing darts at a disk (i.e. a filled in circle) shaped board of radius 5. We assume that the position of the dart is a uniformly chosen point in the disk. The board has a disk shaped bullseye with radius 1. Suppose that we throw a dart 1900 times at the board. Estimate the probability that we hit the bullseye at least 75 times.
A projectile is launched at an angle theta with respect to the surface with velocity v0 (deterministic). If the angle of inclination is a uniform random variable in [0, pi/2 ], calculate the distribution function of the variable R defined as the point of impact of the projectile on the ground, measured from the origin. Also calculate your expected value
Suppose that Y is a continuous random variable. Show EY yfr(y)dy.
Your internal body temperature T in °F is a Gaussian (μ =98.6, σ = 0.4) random variable. In terms of the Φ function, find P[T > 100]. Does this model seem reasonable?
The distribution of a random vector is given by a function:f(x,y) = y/2 - x/2 ; for 0 < x < 1 and 2 < y < 3 = 0 ; otherwise Determine P( X > Y - 2), and sketch the graf.
Suppose claim amounts at a health insurance company are independent of one another. In the first year calim amounts are modeled by a gamma random variable X with alpha=40, and beta=3. In the second year, individual claim amounts are modeled by random variable Y=1.05X+3. Let W be the average of 30 claim amounts in year two set up the equation to model the random variable W. a) Find the moment generating function of W b) Based on moment generating function of W is W also a gamma distribution? if so what are the parameters? c) Find the approximate probability that W is between 125$ and 130$.