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.
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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...
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken the two shorter sides of a right-angled triangle. Let Θ be the smallest angle in this triangle. Define Y-tan Θ and W-cot Θ. Find E(Y) and the pd.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?
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 stick of length L is broken in two pieces at a point which is uniformly...
A stick of length L is broken in two pieces at a point which is uniformly distributed on the stick’s length. What is the expectation of the ratio of the smaller length to the larger?
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...
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...
) 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...
We can play a variation of a child's game called "What am 12" that I will call "Where am 1? (a) I am a square. The intersection of my two diagonals lies at the point (4, 4), and the l...
We can play a variation of a child's game called "What am 12" that I will call "Where am 1? (a) I am a square. The intersection of my two diagonals lies at the point (4, 4), and the length of each of my sides is 8. My sides form horizontal and vertical lines. Where am I? Order your answers from smallest to largest x, then from smallest to largest y.) x, y) (x, y)s x, y) (x, y) (b)...
Suppose that a point is chosen at random on a stick of unit length and that the stick is broken the two shorter sides of a right-angled triangle. Let Θ be the smallest angle in this triangle. Define Y-tan Θ and W-cot Θ. Find E(Y) and the pd.f of W
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 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...
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...
) 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...
We can play a variation of a child's game called "What am 12" that I will call "Where am 1? (a) I am a square. The intersection of my two diagonals lies at the point (4, 4), and the length of each of my sides is 8. My sides form horizontal and vertical lines. Where am I? Order your answers from smallest to largest x, then from smallest to largest y.) x, y) (x, y)s x, y) (x, y) (b)...
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