For each n ∈ N, let f_n(x) = a_nx + b_n where a_n,b_n are
sequences of...
For each n ∈ N, let f_n(x) = a_nx + b_n where a_n,b_n are
sequences of real numbers. Prove that if f_n → f on [0,1] where f
is a function on [0,1], then the convergence is necessarily
uniform.
Solutions
Expert Solution
First we find the function f by using the pointwise convergence
.then we proceed for uniform convergence of sequence fn.
Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an
integer >1.
a. Find the value of the constant C (in terms of n) that makes
this a probability density function.
b. For what values of n does the expected value E(X) exist?
Why?
c. For what values of n does the variance var(X) exist? Why?
using matlab, compute and plot y [n] = x [n]* h [n],
where
a. x [n] = h [n] = a^n (0 <=n <=40) & a = 0.5
b. x [n] = cos [n]; h [n] = u [n]; n = 0:4:360
c. x [n] = sin [n] ; h [n] = a^n; n:4:360; a = 0.9
The question is correct.
Let X be an n-element set of positive integers each of whose
elements is at most (2n - 2)/n. Use the pigeonhole
principle to show that X has 2 distinct nonempty subsets A ≠ B with
the property that the sum of the elements in A is equal to the sum
of the elements in B.
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
(a)
P(15 ≤ X ≤ 20)
p
P(15 ≤ X ≤ 20)
P(14.5 ≤ Normal ≤ 20.5)
0.5
1
2
0.6...
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
P(20 ≤ X)
p
P(20 ≤ X)
P(19.5 ≤ Normal)
0.5
0.6
0.8
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
(a)
P(15 ≤ X ≤ 20)
p
P(15 ≤ X ≤ 20)
P(14.5 ≤ Normal ≤ 20.5)
0.5
0.6
0.8
(B)P(X...
Let A be an m x n matrix and b and x be vectors such that
Ab=x.
a) What vector space is x in?
b) What vector space is b in?
c) Show that x is a linear combination of the columns of A.
d) Let x' be a linear combination of the columns of A. Show that
there is a vector b' so that Ab' = x'.
Let X Geom(p). For positive integers n, k define
P(X = n + k | X > n) = P(X = n + k) / P(X > n) :
Show that P(X = n + k | X > n) = P(X = k) and then briefly
argue, in words, why this is true for geometric random
variables.