Suppose there is a square dartboard that is 8 inches by 8 inches. If you throw 65 darts at the bo...
Suppose there is a square dartboard that is 8 inches by 8 inches. If you throw 65 darts at the board (and hit the board every time), prove that at least 2 of them will be less than 1.5 inches apart. (You may use the fact that the longest distance between two points in a 1 × 1 square is √ 2 ≈ 1.41).
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Suppose there is a square dartboard that is 8 inches by 8 inches. If you throw 65 darts at the bo...
Suppose you throw a dart at a circular target of radius 5 inches. Assuming that you...
Suppose you throw a dart at a circular target of radius 5 inches. Assuming that you hit the target and that the coordinates of the outcomes are chosen at random. Find the probability that the dart falls within the first quadrant and within 2 inches of the rim.
Suppose you want to throw a dart across a room at a wall-mounted target and hit...
Suppose you want to throw a dart across a room at a wall-mounted
target and hit the target exactly at its center point. We call this
center point of the target its “bull’s-eye”. The bull’s-eye is a
horizontal distance d from the dart’s point of release and a height
habove the dart’s point of release. To hit the bull’s-eye, you have
a choice of the speed v at which you throw the dart and a choice of
the angle θ...
1. Bull's-Eye Bonanza. (Allotted Time 45 minutes) Suppose you want to throw a dart across a...
1. Bull's-Eye Bonanza. (Allotted Time 45 minutes) Suppose you want to throw a dart across a room at a wall-mounted target and hit the target exactly at its center point. We call this center point of the target its "bull's eye". The bull's-eye is a horizontal distance d from the dart's point of release and a height h above the dart's point of release. To hit the bul'seye, you have a choice of the speed s at which you throw...
1. Suppose that you do not know your true success rate at basketball free throws. Out...
1. Suppose that you do not know your true success rate at basketball free throws. Out of the 17 attempts, you were able to make 9 of them. a) What is the point estimate associated to the true probability of success p? (4 decimals) b) What is the standard error associated to this estimate? (4 decimals) 2. Suppose that you claimed to someone that you can make basketball free-throw shots 80% of the time. Someone didn't believe you with a level of...
1a) Suppose you flip a weighted coin 19 times with probability of heads 0.55. Let X...
1a) Suppose you flip a weighted coin 19 times with probability of heads 0.55. Let X equal the number of heads. What is the mean number of heads (to two decimal places)? 1b) Suppose we have weighted coin with probability of heads being 0.30. What is the probability of getting four heads in five flips, to 2 decimal places? 2) Suppose a particular stock follows the "random walk" model where on any given day there is a 0.02 chance that...
1.9.1 Exercise 1 Describe three decision-making situations that you face at least once a year in...
1.9.1 Exercise 1 Describe three decision-making situations that you face at least once a year in your life and that involve uncertainty. Describe the sources of uncertainty and your method to come to a decision. For example, you may have to arrive at a meeting on Mondays at 5 pm, so you have to decide when you will start your trip towards the meeting place. сеа 1.9.2 Exercise 2 Goal: Do many random experiments and estimations. Identify the elements involved...
1-1 Suppose you want to make a scale model of a hydrogen atom. You choose, for...
1-1 Suppose you want to make a scale model of a hydrogen atom. You choose, for the nucleus, a small ball bearing with a radius of 1.5 mm. The radius of the hydrogen atom is 0.529 × 10−10 m and the radius of the nucleus is 1.2 × 10−15 m. (a) What would be the radius of the model? (b) Suppose that now you want to make a scale model of the solar system using the same ball bearing as...
As a general comment, remember that showing two random variables have the same CDFor PDF is...
As a general comment, remember that showing two random variables have the same CDFor PDF is sufficient for showing that they have the same distribution. a) First, let us see how to generate an exponential random variable with a uniform random vari- (b) Let M.N2 ~ y (0, î ), where Ni and N2 are independent. Prove that NỈ + N: ~ Expo( 1 /2). able. Let U1~Uniform(0, 1). Prove that-InU1Expo(1). Hint: You may use the fact that over a...
Question 1 (50 pts): Suppose that a client of yours measure the heights (in inches) of...
Question 1 (50 pts): Suppose that a client of yours measure the heights (in inches) of n - 30 wheats grown at locations of various elevations (measured as meters above sea levels). Af- ter some discussion, you decided to fit a linear regression of wheat heights (denoted as yi) on the elevations of the locations (denoted as zi) as follows where ei, E2, . . . , En are i.i.d. errors with Elei] 0 and var(G) σ2. You calculated some...
SECTION 3.5: EXERCISES 1.) Suppose you want to construct a rectangular box with no lid that...
SECTION 3.5: EXERCISES 1.) Suppose you want to construct a rectangular box with no lid that is to hold 580 cubic inches of volume. The cost of the material for the sides is $0.10 per square inch, while the cost of the material for the base (made of a sturdier and more expensive material) is $0.25 per square inch. If the base is to have a length that is twice the width, what dimensions will minimize the total cost of...
Suppose you want to throw a dart across a room at a wall-mounted
target and hit the target exactly at its center point. We call this
center point of the target its “bull’s-eye”. The bull’s-eye is a
horizontal distance d from the dart’s point of release and a height
habove the dart’s point of release. To hit the bull’s-eye, you have
a choice of the speed v at which you throw the dart and a choice of
the angle θ...
1. Bull's-Eye Bonanza. (Allotted Time 45 minutes) Suppose you want to throw a dart across a room at a wall-mounted target and hit the target exactly at its center point. We call this center point of the target its "bull's eye". The bull's-eye is a horizontal distance d from the dart's point of release and a height h above the dart's point of release. To hit the bul'seye, you have a choice of the speed s at which you throw...
1.9.1 Exercise 1 Describe three decision-making situations that you face at least once a year in your life and that involve uncertainty. Describe the sources of uncertainty and your method to come to a decision. For example, you may have to arrive at a meeting on Mondays at 5 pm, so you have to decide when you will start your trip towards the meeting place. сеа 1.9.2 Exercise 2 Goal: Do many random experiments and estimations. Identify the elements involved...
As a general comment, remember that showing two random variables have the same CDFor PDF is sufficient for showing that they have the same distribution. a) First, let us see how to generate an exponential random variable with a uniform random vari- (b) Let M.N2 ~ y (0, î ), where Ni and N2 are independent. Prove that NỈ + N: ~ Expo( 1 /2). able. Let U1~Uniform(0, 1). Prove that-InU1Expo(1). Hint: You may use the fact that over a...
Question 1 (50 pts): Suppose that a client of yours measure the heights (in inches) of n - 30 wheats grown at locations of various elevations (measured as meters above sea levels). Af- ter some discussion, you decided to fit a linear regression of wheat heights (denoted as yi) on the elevations of the locations (denoted as zi) as follows where ei, E2, . . . , En are i.i.d. errors with Elei] 0 and var(G) σ2. You calculated some...
SECTION 3.5: EXERCISES 1.) Suppose you want to construct a rectangular box with no lid that is to hold 580 cubic inches of volume. The cost of the material for the sides is $0.10 per square inch, while the cost of the material for the base (made of a sturdier and more expensive material) is $0.25 per square inch. If the base is to have a length that is twice the width, what dimensions will minimize the total cost of...
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