Homework Answers

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.

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.

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.

Determine whether the following functions are continuous on the given interval. Show your complete solution.

1. ƒ(𝑥)=√4−𝑥2 ;[−2,2]

2. ƒ(𝑥)=3𝑥2 −𝑥+5 ; (−∞,+∞)

Determine whether the following functions are continuous at a given point. Show your complete solution.

1. ƒ(𝑥)=3𝑥2−4𝑥+2at𝑥=2

2. ƒ(𝑥)=𝑥2−6𝑥−3at𝑥=4