Question

1) Give an example for each of the following:

a) A random experiment with finite space of elementary events.

b) A random experiment with infinite countable space of elementary events.

c) A random experiment with continuous space of elementary events.

Answer #1

Sample space S of a random experiment is defined as the set of
all

possible outcomes.

A sample space can be finite, countably infinite or uncountably
infinite.

1. Toss a coin two times

S1 = {(H, H), (H, T), (T, H), (T, T)}

S1 is countable, S1 is called a discrete sample space. Define

B = {H, T}, then S1 = B × B.

2. Toss a dice until a ‘six’ appears and count the number
of

times the dice was tossed.

S2 = {1, 2, 3, …};

S2 is discrete and countably infinite (one-to-one
correspondence

with positive integers)

3. Pick a number X at random between zero and one, then pick
a

number Y at random between zero and X.

S3 = {(x, y): 0 ≤ y ≤ x ≤ 1};

S3 is a continuous sample space.

For each of the following, give a specific example of
sets that satisfy the stated conditions.
(a) A and B are infinite and |B −A| = 3.
(b) A is infinite, U is infinite, and Ac is infinite. (c) A is
infinite, U is infinite, and Ac is finite.

Give an example of a space that is separable, but not 2nd-
countable. Prove that the space you give is separable and prove it
is not 2nd- countable

A random experiment has sample space S = {a, b, c, d}. Suppose
that P({c, d}) = 3/8, P({b, c}) = 6/8, and P({d}) = 1/8. Use the
axioms of probability to find the probabilities of the elementary
events.

Give an example of an experiment where the sample space does
NOT have that each outcome is equally likely.
Explain your reasoning as best you can - why are some outcomes more
likely than others?

Prove for each:
a. Proposition: If A is finite and B is countable, then A ∪ B is
countable.
b. Proposition: Every subset A ⊆ N is finite or countable.
[Similarly if A ⊆ B with B countable.]
c. Proposition: If N → A is a surjection, then A is finite or
countable. [Or if countable B → A surjection.]

give an example of a finite dimensional real inner product space
V, an operator T on V and a subspace W of V such that W is a
T-invariant subspace of V. is it possible to find such an example
such that the operator T is self-adjoint?

Discrete Random Variables have either a finite or countable
number of values.
True
False
An example of continuous variables is bushels of wheat per
acre.
True
False
The Mean Value of a discrete probability distribution (denoted
by mu is a weighted average of the x-values AND represents the
average values of all possible outcomes.
True
False
Explain why in a binomial probability distribution, p + q =1.
Make one sentence work.

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

Give an example with a proof of an infinite-dimensional vector
space over R

Define each of the following terms and give an example to
illustrate. a. Line spectra: b. Continuous spectra: c. Quantized:
d. Transition

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