Let (?,?,?) be a probability space and suppose that ?∈? is an
event with ?(?)>0.
Prove...
Let (?,?,?) be a probability space and suppose that ?∈? is an
event with ?(?)>0.
Prove that the function ?:?→[0,1] defined by ?(?)=?(?|?) is a
probability on (?,?).
Let (Ω, F , P) be a probability space. Suppose that Ω is the
collection of all possible outcomes of a single iteration of a
certain experiment. Also suppose that, for each C ∈ F, the
probability that the outcome of this experiment is contained in C
is P(C).
Consider events A, B ∈ F with P(A) + P(B) > 0. Suppose that the
experiment is iterated indefinitely, with each iteration identical
and independent of all the other iterations, until...
Suppose my sample space consists of current STATS125
students.
Let T be the event that a randomly selected current STATS125
student usually attends Tuesday lectures,
W be the event that a randomly selected current STATS125 student
usually attends Wednesday lectures
H be the event that a randomly selected current STATS125 student
usually attends Thursday lectures.
a) Draw 4 Venn diagrams and use shading to identify the
following sets:
i. T ? H
ii. T^complement ? (W ? H)
iii. the...
#18
Events A and B are mutually exclusive. Suppose event occurs
with probability 0.05 and event occurs with probability 0.36. a.
Compute the probability that B occurs or does not occur (or both).
b. Compute the probability that either occurs without occurring or
occurs without occurring. (If necessary, consult a list of
formulas.) ?
#19
Events and are mutually exclusive . Suppose event occurs with
probability 0.11 and event B occurs with probability 0.81. If does
not occur, what is...
Exercise 4 (Indicator variables). Let (Ω, P) be a probability
space and let E ⊆ Ω be an event. The indicator variable of the
event E, which is denoted by 1E , is the RV such that 1E (ω) = 1 if
ω ∈ E and 1E(ω)=0ifω∈Ec.Showthat1E
isaBernoullivariablewithsuccessprobabilityp=P(E).
Exercise 5 (Variance as minimum MSE). Let X be a RV. Let xˆ ∈ R
be a number, which we consider as a ‘guess’ (or ‘estimator’ in
Statistics) of X . Let...
1. The probability of event A is 60%, the probability of event B
is 40%, and the probability of either A or B is 65%.
What is the probability of events A and B simutaneously?
2. 50% of items are Type A, 30% are Type B, and 20% are Type C.
each type is broken up into V1 and V2. Type A is 80% V1, Type B is
60% V1, and Type C is 30% V1.
If a randomly selected...
Prove the following statements!
1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r
when 24|(k−r). If g : S→S is defined by
(a) g(m) = f(7m) then g is injective and
(b) g(m) = f(15m) then g is not injective.
2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is
injective.
3. Let f : A→B and g : B→C be surjective....