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) 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
Determine whether or not the following are continuous functions.
1. ƒ(𝑥)=5𝑥+3 2. ƒ(𝑥)=−4𝑥+2
3. ƒ(𝑥)=2𝑥2 +𝑥−3
4. ƒ(𝑥)={2𝑥−3 iƒ 𝑥≥2 −2𝑥 + 2 iƒ 𝑥 < 2
5. ƒ(𝑥)={|𝑥+2|iƒ𝑥G−2 4 iƒ 𝑥 = −2
Evaluate the following limits.
1. limX→+∞ 5X+6
2. limX→−∞ e4X+7
3.limX→−∞(0.2)X
How many ways can 4 baseball players and 3 basketball players be selected from 12 baseball players and 9 basketball players?
A study by Thienprasiddhi et al. (A-4) examined a sample of 16 subjects with open-angle glaucoma
and unilateral hemifield defects. The ages (years) of the subjects were:
62 62 68 48 51 60 51 57
57 41 62 50 53 34 62 61
Can we conclude that the mean age of the population from which the sample may be presumed to have been drawn is less than 60 years? Let alpha = 0.5
