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?
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A stick of length L is broken in two pieces at a point which is
uniformly...
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?
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 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...
(1 point) Two points along a straight stick of length 49 cm are randomly selected. The...
(1 point) Two points along a straight stick of length 49 cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least 7.5 cm. probability =
(1 point) Two points along a straight stick of length 38 cm are randomly selected. The...
(1 point) Two points along a straight stick of length 38 cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least 4.5 cm. probability =
A piece of lumber of fixed length L is broken into two sub-pieces at position X...
A piece of lumber of fixed length L is broken into two sub-pieces at position X ~ U(0, L). Let Y denote the length of the shorter sub-piece. a) Find the cumulative distribution function of Y. Hint: First express Y as a piecewise function in terms of X. b) Find the probability distribution function of Y. Identify the name of the distribution that Y follows. Also, write down the parameter value(s) of this distribution.
5 (10 points). We start with a stick of length L. Break it at a point...
5 (10 points). We start with a stick of length L. Break it at a point cho sen uniformly randomly and keep the piece that contains the left end. Let its length be Y. Repeat the same process on this stick with length Y. Let X be the length of the remaining piece. a) (2 points) Find the joint PDF of X and Y b) (3 points) Find the marginal PDF of X c) (3 points) Use the PDF of...
A piece of lumber of fixed length ?? is broken into two sub pieces at position...
A piece of lumber of fixed length ?? is broken into two sub pieces at position ?? ~ ??(0,??). Let ?? denote the length of the shorter sub piece. a. Find the cumulative distribution function of ??. Hint: First express ?? as a piecewise function in terms of ??. b. Find the probability distribution function of ??. Identify the name of the distribution that ?? follows. Also write down the parameter value(s) of this distribution.
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
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,...
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 point) Two points along a straight stick of length 38 cm are randomly selected. The stick is then broken at those two points. Find the probability that all of the resulting pieces have lenght at least 4.5 cm. probability =
5 (10 points). We start with a stick of length L. Break it at a point cho sen uniformly randomly and keep the piece that contains the left end. Let its length be Y. Repeat the same process on this stick with length Y. Let X be the length of the remaining piece. a) (2 points) Find the joint PDF of X and Y b) (3 points) Find the marginal PDF of X c) (3 points) Use the PDF of...
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
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,...
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