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Suppose that a point is chosen at random on a stick of unit length and that...
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Then, we form a right angle with two pieces of stick, forming the two sho...
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken into two pieces at that point. Then, we form a right angle with two pieces of stick, forming the two shorter sides of a right-angled triangle. Let Θ be the smallest angle in this triangle. Define Y = tanΘ and W = cotΘ. Find E(Y ) and the p.d.f of W.
A stick of length one is broken into two pieces at a random point. What is...
A stick of length one is broken into two pieces at a random point. What is the probability that the length of the longer piece will be at least three times the length of the shorter piece?
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the pr...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
A stick is broken into three pieces at two randomly chosen points on the stick. What is the proba...
A stick is broken into three pieces at two randomly chosen points on the stick. What is the probability that no piece is longer than half the length of the stick? To do this problem, it is useful to split it into the following steps (assuming the length of stick is 1). (a) Let U1 and U2 are independent and uniformly distributed on (0, 1). Define X = min(U1, U2) and Y = max(U1, U2). Use the fact that P(a...
A point is chosen at random on a line segment of length 12. Find the probability...
A point is chosen at random on a line segment of length 12. Find the probability that the ratio of the shorter to the longer segment is less than 3/20.
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then...
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X. Uniform (a,...
Find the lengths of the missing sides if side ais opposite angle A, side bis opposite...
Find the lengths of the missing sides if side ais opposite angle A, side bis opposite angle B, and side c is the hypotenuse Draw a right triangle labeling side a, side b, and hypotenuse, and their corresponding angles. Define the ratio for tan(A) in terms of ab, or what proportion relates the given valu for tan(A) and the defined ratio for tan(A)? How is this proportion solved for a with values for side and sideb, use the Pythagorean Theorem...
) 8. Suppose a triangle is constructed where two sides have fixed length a and b, but the third side has variable length x You can imagine there is a pivot point where the sides of fixed length a and...
) 8. Suppose a triangle is constructed where two sides have fixed length a and b, but the third side has variable length x You can imagine there is a pivot point where the sides of fixed length a and b meet, forming an angle of θ. By changing the angle θ, the opposite side will either stretch or contract (a) Let K(x)- Vs(s - a)(s -b)(s - x), where s is the semiperimeter of the triangle. Accord ing to...
3. Suppose a value is chosen "at random" in the interval [0,6]. In other words, r...
3. Suppose a value is chosen "at random" in the interval [0,6]. In other words, r is an observed value of a random variable X U(0,6). The random variable X divides the interval [0,6] into two subintervals, the lengths of which are X and 6- X respectively. Denote by Y min(X, 6-X), the length of the shorter one of the two intervals. Find e probability PY> ) for any given y. Then find both the cdf and the pdf of...
A stick of length one is broken into two pieces at a random point. What is the probability that the length of the longer piece will be at least three times the length of the shorter piece?
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
4. Uniform Stick-Breaking A point X is chosen uniformly from the interval (0, 10) and then a point Y is chosen uniformly from the interval (0, X). This can be imagined as snapping a stick of length 10 and then snapping one of the broken bits. Such processes are called stick-breaking processes. a) Find E(X) and Var(X). See Section 15.3 of the textbook for the variance of the uniform. b) Find E(Y) and Var(Y) by conditioning on X. Uniform (a,...
Find the lengths of the missing sides if side ais opposite angle A, side bis opposite angle B, and side c is the hypotenuse Draw a right triangle labeling side a, side b, and hypotenuse, and their corresponding angles. Define the ratio for tan(A) in terms of ab, or what proportion relates the given valu for tan(A) and the defined ratio for tan(A)? How is this proportion solved for a with values for side and sideb, use the Pythagorean Theorem...
) 8. Suppose a triangle is constructed where two sides have fixed length a and b, but the third side has variable length x You can imagine there is a pivot point where the sides of fixed length a and b meet, forming an angle of θ. By changing the angle θ, the opposite side will either stretch or contract (a) Let K(x)- Vs(s - a)(s -b)(s - x), where s is the semiperimeter of the triangle. Accord ing to...
3. Suppose a value is chosen "at random" in the interval [0,6]. In other words, r is an observed value of a random variable X U(0,6). The random variable X divides the interval [0,6] into two subintervals, the lengths of which are X and 6- X respectively. Denote by Y min(X, 6-X), the length of the shorter one of the two intervals. Find e probability PY> ) for any given y. Then find both the cdf and the pdf of...
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