Search & Filtering
Determine whether the functions ƒ(𝑥) = √4−𝑥2 is continuous on the interval
[−4,4]. Show your complete solution.
Determine whether the following functions are continuous at a given point. Show your complete solution.
1. ƒ(𝑥) = 𝑥2−4 at 𝑥 = 2 𝑥−2
2. ƒ(𝑥) = 𝑥2−25 at 𝑥 = 2 𝑥−5
Suppose that a random variable X has a Poisson distribution with parameter λ. The
parameter λ itself is a random variable with the exponential distribution with mean 1
c ,
where c is a constant. Show that
P(X = k) =
c
(c + 1)k+1
) Prove that for any discrete bivariate random variable (X, N) for which the first
moments of X and N exists,
E(X) = E [E (X|N)]
(b) The number N of customers entering the University of Ghana book-shop each day
is a random variable. Suppose that each customer has, independently of other
customers, a probability θ of buying at least one book. Let X denote the number
of customers that buy at least one book each day.
Describe without proof the distribution of X conditional on N = n. Hence use the
results in (a) to evaluate the expectation of X if N has the distribution.
i. P(N = k) = M
k θk
(1 − θ)M−k
, k = 0, 1, · · · , M
ii. P(N = k) = θ(1 − θ)k
, k = 0, 1, 2, · · · ,
iii. P(N = k) =
e−θθk
k!
, k = 0, 1, 2, · · ·
iv. P(N = k) = θ(1 − θ)k−1
, k = 1, 2, · · ·
Find the probability distribution of X if N has the distribution in (b) i-iv.
A6. Let X and Y be two independent, nonnegative integer-valued random variables whose
distribution has the property
P (X = x|X + Y = x + y) =
m
x
n
y
m+n
x+y
for all nonnegative integers x and y where m and n are given positive integers. Assume
that P(X = 0) and P(Y = 0) are strictly positive. Show that both X and Y have
binomial distributions with the same parameter p, the other parameters being m and
n respectively.
Let Ω be a sample space of rolling a fair die once and observing the face value. Let A
be the event with element 1 and 3. Show that the smallest σ − f ield containing A is
a σ − f ield.
Let F and G be two sigma-fields on Ω. Prove that F ∩ G is also a sigma-field on Ω.
Show by example that F ∪ G may fail to be sigma-field if Ω = {1, 2, 3}.
A3. Let (Ω, F, P) be a probability space and let H ∈ F with P(H) > 0. For any arbitrary
A ∈ F, let
PH(A) =
P(A ∩ H)
P(H)
Show that (Ω.F, PH) is a probability space.
Suppose F is a σ − algebra of subsets of Ω and let B ∈ F.
(a) Show that {A ∩ B : A ∈ F} is a σ − algebra of subsets of B
(b) If the sum of possible outcomes of a fair die thrown twice is ω and the sample
space Ω = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and B = {2, 4, 6, 8, 10} ∈ F, then deduce
that for any subset A of Ω, {A ∩ B : A ∈ F} is σ − algbra of subset of B
In an experiment of tossing a fair coin four times. Let the sample space Ω be the
number of tails observed and ϕ be the impossible event.
(a) List the Ω and Sigma field F, with the maximum cardinality.
(b) If A1, A2, A3, A4 are subsets of Ω, show that the class of sets F = {ϕ, A1, A2, A3, A4, Ω}
is a σ − f ield.
(c) If P is a function defined on F, what properties must P satisfy for the triple
(Ω, F, P) to be called a probability space.
#339914 on Dec 2023