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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...
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...
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?
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.
)on 4. Suppose X and y are continuous random variables with joint density funstion the unit...
)on 4. Suppose X and y are continuous random variables with joint density funstion the unit square [0, 1] x [0, 1]. (a) Let F(r,y) be the joint CDF. Compute F(1/2, 1/2). Compute F(z,y). (b) Compute the marginal densities for X and Y (c) Are X and Y independent? (d) Compute E(X), E(Y), Cov(X,y)
Exercise about two-dimensional random variables, independence and covariation: Suppose, two-dimensional random variable...
Exercise about two-dimensional random variables, independence
and covariation:
Suppose, two-dimensional random variable (X, Y) has probability density function as follows: 0y1 + f(x, y) 2xy) ,0 <x<1, otherwise 0 Find c Find marginal probability density functions of X and Y-find f(x) and f(y) and find if X and Y are independent; Find joint (X, Y) distribution function; Find covariation of X and Y find Cov(X, Y) and correlation p(X, Y). What can be concluded?
Suppose, two-dimensional random variable (X, Y)...
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
We said in class that two events A and B are indep(ndent if μ(An B) 6....
We said in class that two events A and B are indep(ndent if μ(An B) 6. μ(A)a(B). Sinilarly, two random variables X and Y are said to be independent if their joint density fx.y(r,y) can be expressed as the product of the marginal densities fx(x)fv(y). Let X and Y be independent (scalar) random variables, and ZX Y be a new random variable defined as the sum of X and Y. Show that the moment generating function mz(t) of Z is...
suppose that two random variables X and Y have joint Pdf, f(x,y) = 1/√2π * e^(-((x2y2)/2)-y)...
suppose that two random variables X and Y have joint Pdf, f(x,y) = 1/√2π * e^(-((x2y2)/2)-y) * y2 -inf < x < inf and y > 0 a) Are X and Y independent? Justify. b) Find the distribution of Y
4. Two numbers, and y, are selected at random from the unit interval [0, 1. Find...
4. Two numbers, and y, are selected at random from the unit interval [0, 1. Find the probability that each of the three line segements so formed have length > . For example, if 0.3 and y0.6, the three line segements are (0,0.3), (0.3,0.6) and (0.6, 1.0). All three of these line segments have length >. HINT: you can solve this problem geometrically. Draw a graph of and y, with x on the horizontal axis and y on the vertical...
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...
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?
)on 4. Suppose X and y are continuous random variables with joint density funstion the unit square [0, 1] x [0, 1]. (a) Let F(r,y) be the joint CDF. Compute F(1/2, 1/2). Compute F(z,y). (b) Compute the marginal densities for X and Y (c) Are X and Y independent? (d) Compute E(X), E(Y), Cov(X,y)
Exercise about two-dimensional random variables, independence
and covariation:
Suppose, two-dimensional random variable (X, Y) has probability density function as follows: 0y1 + f(x, y) 2xy) ,0 <x<1, otherwise 0 Find c Find marginal probability density functions of X and Y-find f(x) and f(y) and find if X and Y are independent; Find joint (X, Y) distribution function; Find covariation of X and Y find Cov(X, Y) and correlation p(X, Y). What can be concluded?
Suppose, two-dimensional random variable (X, Y)...
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
We said in class that two events A and B are indep(ndent if μ(An B) 6. μ(A)a(B). Sinilarly, two random variables X and Y are said to be independent if their joint density fx.y(r,y) can be expressed as the product of the marginal densities fx(x)fv(y). Let X and Y be independent (scalar) random variables, and ZX Y be a new random variable defined as the sum of X and Y. Show that the moment generating function mz(t) of Z is...
4. Two numbers, and y, are selected at random from the unit interval [0, 1. Find the probability that each of the three line segements so formed have length > . For example, if 0.3 and y0.6, the three line segements are (0,0.3), (0.3,0.6) and (0.6, 1.0). All three of these line segments have length >. HINT: you can solve this problem geometrically. Draw a graph of and y, with x on the horizontal axis and y on the vertical...
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