Question

Let n greater than or equal to 1 be a positive integers, and let
X_{1}, X_{2},....., X_{n} be closed subsets
of R. Show that X_{1} U X_{2} U ... X_{n}
is also closed.

Answer #1

Let n ≥ 2 and x1, x2, ..., xn > 0 be such that x1 + x2 + · ·
· + xn = 1. Prove that √ x1 + √ x2 + · · · + √ xn /√ n − 1 ≤ x1/ √
1 − x1 + x2/ √ 1 − x2 + · · · + xn/ √ 1 − xn

Suppose {x1,x2,...xn} is linear dependent and {x1,x2} is linear
independent.
Show that: 2 < than or equal to
(Span(x1,x2,...xn)) < n.

1. Let n be an odd positive integer. Consider a list of n
consecutive integers.
Show that the average is the middle number (that is the number
in the
middle of the list when they are arranged in an increasing
order). What
is the average when n is an even positive integer instead?
2.
Let x1,x2,...,xn be a list of numbers, and let ¯ x be the
average of the list.Which of the following
statements must be true? There might...

Consider three positive integers, x1, x2, x3, which satisfy the
inequality below:
x1 + x2 + x3 = 17.
Let’s assume each element in the sample space (consisting of
solution vectors (x1, x2, x3) satisfying the above conditions) is
equally likely to occur. For example, we have equal chances to have
(x1, x2, x3) = (1, 1, 15) or (x1, x2, x3) = (1, 2, 14). What is the
probability the events x1 + x2 ≤ 8 occurs, i.e., P(x1...

Consider the sequence (xn)n given by x1 = 2, x2 = 2 and xn+1 =
2(xn + xn−1).
(a) Let u, w be the solutions of the equation x 2 −2x−2 = 0, so
that x 2 −2x−2 = (x−u)(x−w). Show that u + w = 2 and uw = −2.
(b) Possibly using (a) to aid your calculations, show that xn =
u^n + w^n .

Let X1, X2, . . . , Xn be a random sample of size n from a
distribution with variance σ^2. Let S^2 be the sample variance.
Show that E(S^2)=σ^2.

Let x1, x2, ..., xk be linearly independent vectors in R n and
let A be a nonsingular n × n matrix. Define yi = Axi for i = 1, 2,
..., k. Show that y1, y2, ..., yk are linearly independent.

Let X1, X2, ..., Xn be a random sample (of size n) from U(0,θ).
Let Yn be the maximum of X1, X2, ..., Xn.
(a) Give the pdf of Yn.
(b) Find the mean of Yn.
(c) One estimator of θ that has been proposed is Yn. You may
note from your answer to part (b) that Yn is a biased estimator of
θ. However, cYn is unbiased for some constant c. Determine c.
(d) Find the variance of cYn,...

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications pleasw, thank you.

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications please, thank you.

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