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.
A piece of lumber of fixed length L is broken into two sub-pieces at position X...
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.
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. A piece of wire of length L cm is cut into two pieces. One piece, of length x cm, is made into a circle; the rest is made into a square. Denote by A(x) the sum of the area of the circle and the area of the square. (a) (5 points) Find I'm which makes A(z) MINIMUM, and find A(1m). (b) (5 points) Find xm which makes A(x) MAXIMUM, and find A(zm).
8. A piece of wire of length L cm is cut into two pieces. One piece, of length x cm, is made into a circle; the rest is made into a square. Denote by A(z) the sum of the area of the circle and the area of the square. (a) (5 points) Find Im which makes A(z) MINIMUM, and find A(I'm). (b) (5 points) Find am which makes A(z) MAXIMUM, and find A(IM).
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...
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 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...
Suppose density function positively valued continuous random variable X has the probability a fx(x)kexp 20 fixed 0> 0 for 0 o0, some k > 0 and for (a) Find k such that f(x) satisfies the conditions for a probability density function (4 marks) (b) Derive expressions for E[X] and Var[X (c) Express the cumulative distribution function Fx(r) in terms of P(), the stan dard Normal cumulative distribution function (8 marks) (8 marks) (al) Derive the probability density function of Y...
Question. 4 (20%) A uniformly loaded beam of length "L" is supported at both ends. The deflection y(x) is a function of horizontal position x and is given by the differential equation on dEl d1 Beat dE 4() Assume q(x) is constant. Determine the equation for y(x) in terms of different variables. Hint: Use laplace transform. Below are boundary conditions: (L)ono dene y"(o) o no deflection at x= 0 and L no bending moment at x 0 and L y...
Consider a system consisting of three components as pictured. The system will continue to function as long as the first component functions and either component 2 or component 3 functions. Let X1, X2, and X3 denote the lifetimes of components 1, 2, and 3, respectively. Suppose the Xi's are independent of one another and each X, has an exponential distribution with parameter λ. (a) Let Y denote the system lifetime. Obtain the cumulative distribution function of Y and differentiate to...