Download Probability 2. 3. Homework 2010.03.21.

Probability 2.
3. Homework
2010.03.21.
Moment generating function, generating function, characteristic function
1. Determine the moment generating function and the characteristic function of the distributions given below! By the help of these, calculate the expectation of the corresponding
random variables! BIN(𝑛, 𝑝), POI(𝜆), GEO(𝑝), NBIO(𝑟, 𝑝), DU[0, 𝑛].
(Here NBIO(𝑟, 𝑝) denotes the negative binomial distribution, DU[0, 𝑛] denotes the discreet uniform distribution on the 𝑛 + 1 numbers.)
2. Let 𝑋 be an ℕ-valued random variable with generating function 𝑃 (𝑧). Determine the
generating function of the random variables 𝑌 := 𝑋 + 1 and 𝑍 := 2𝑋! Also determine
the generating function of the sequences 𝑎𝑛 := P(𝑋 ≤ 𝑛), 𝑏𝑛 := P(𝑋 < 𝑛), 𝑐𝑛 :=
P(𝑋 ≥ 𝑛), 𝑑𝑛 := P(𝑋 > 𝑛 + 1), 𝑒𝑛 := P(𝑋 = 2𝑛)! (Mind that these are not
probability distributions).
3. Determine the generating function of the 𝐺𝐸𝑂(𝑝) distribution by conditional recursion
and using the fact that it is memoryless.
4. Determine the moment generating function and the characteristic function of the distributions given below! By the help of these, calculate the expectation of the corresponding
random variables. E[𝑎, 𝑏], EXP(𝜆), GAMMA(𝑠, 𝜆), N(𝜇, 𝜎 2 ).
5. Are the functions below generating functions of some probability distribution? (If yes,
what is the distribution?)
(
)
𝑧−1
(𝑧 + 1)6
2
2
(a) exp
, 𝜆 > 0;
(b)
;
(c)
;
(d)
.
𝜆
64
2−𝑧
1+𝑧
6. Let 𝑋 be a random variable of truncated Poisson distribution:
P(𝑥 = 𝑘) =
𝜆𝑘 𝑒−𝜆
𝑘! 1 − 𝑒−𝜆
(𝑘 = 1, 2, . . .)
Determine the generating function of 𝑋, and by the help of it, calculate E(𝑋) and
D(𝑋)!
7. 𝑋1 and 𝑋2 are independent GEO(𝑝1 ) and GEO(𝑝2 ) distributed random variables. Determine the generating function of 𝑌 = min{𝑋1 , 𝑋2 }!
8. 𝑋1 and 𝑋2 are independent EXP(𝜆1 ) and EXP(𝜆2 ) distributed random variables. Determine the moment generating function of 𝑌 = min{𝑋1 , 𝑋2 }!
9. Random Sum of Random Variables
(a) 𝑋1 , 𝑋2 , . . . are independent identically distributed (i.i.d.) ℤ+ -valued random variables. Furthermore, 𝜈 is a ℤ+ -valued random variable, independent of the 𝑋𝑖 -s.
Determine the generating function of 𝑆𝜈 , where
𝑆𝑛 = 𝑋1 + 𝑋2 + . . . + 𝑋𝑛
1
(𝑛 ≥ 0).
(b) 𝑋1 , 𝑋2 , . . . are i.i.d. random variables, 𝜈 is a ℤ+ -valued random variable, independent of the 𝑋𝑖 -s. Determine the characteristic function and the moment generating
function of 𝑆𝜈 , where
𝑆𝑛 = 𝑋1 + 𝑋2 + . . . + 𝑋𝑛
(𝑛 ≥ 0).
10. Infinitely divisible distributions
(a) We say that a random variable is infinitely divisible if ∀𝑛 ∈ ℕ there exist i.i.d.
random variables 𝑌1𝑛 , 𝑌2𝑛 . . . 𝑌𝑛𝑛 such that
𝑛
∑
𝑌𝑖𝑛 ∼ 𝑋
𝑖=1
Are the distributions 𝐵𝐼𝑁 (𝑛, 𝑝) and 𝑃 𝑂𝐼(𝜆), 𝑁 (𝜇, 𝜎) infinitely divisible?
𝜈
∑
(b) Show that if 𝑋𝑖 i.i.d, 𝜈 ∼ 𝑃 𝑂𝐼(𝜆), then 𝑌 =
𝑋𝑖 is infinitely divisible.
𝑖=1
(c) Show that for ∀ 0 < 𝑝 < 1 there is a probability distribution 𝑞 = (𝑞1 , 𝑞2 , . . . ) and
𝜆 > 0 such that for the i.i.d. sequence 𝑋𝑖 of distribution 𝑞 and 𝜈 ∼ 𝑃 𝑂𝐼(𝜆)
𝜈
∑
𝑋𝑖 ∼ 𝐺𝐸𝑂(𝑝),
𝑖=1
where 𝐺𝐸𝑂(𝑝) denotes the pessimistic geometric distribution, i.e. P(𝑋 = 𝑘) =
(1 − 𝑝)𝑘 𝑝. Using this fact, show that the geometric distribution is infinitely divisible. How can you define the 𝑌𝑖𝑛 -s easily to show that the optimistic geometric
distribution is still infinitely divisible?
1. We throw a dice till we first manage to throw two 6-s consecutively. Let us denote the
number of throws needed by 𝜈! Calculate the generating function of 𝜈 using conditional
recursion, and by the help of this, the expected value and variance of 𝜈.
2. 𝑎𝑛 denotes the number of those dice-throw-sequences (of any length), where the sum of
the points thrown equals 𝑛. Show that the generating function of the sequence {𝑎𝑛 } is
[1 − 𝑧 − 𝑧 2 − . . . − 𝑧 6 ]−1 .
We play the following game on an infinite boardgame-table: we start our walker from
the origin and in each step we throw a dice to move forward. Let us denote by 𝑏𝑛 the
probability that we ever step on the number 𝑛. Calculate the generating function of the
sequence 𝑏𝑛 ! What is the limit of it? (the answer is NOT 1/6....)
3. 𝑌 is a U(0, 1)-distributed random variable. Suppose given 𝑌 = 𝑝, the conditional
distribution of the random variable 𝑋 is BIN(𝑛, 𝑝). Show that the distribution of 𝑋 is
UD[0, 𝑛].
4. Let us denote the vertices of an equilateral triangle by 𝐴, 𝐵, 𝐶! We start a simple
(symmetric) random walk from vertex 𝐴 on the graph and we denote the probabilities
of being at vertices 𝐴, 𝐵, and 𝐶 after 𝑛 steps by 𝑎𝑛 , 𝑏𝑛 and 𝑐𝑛 , respectively. Calculate
the generating functions of these sequences and show that each of the sequences tend
to 1/3!
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5. Let 𝑌 be a simple random variable with moment generating function 𝑀 (𝑡). Show that
𝑃 (𝑌 > 0) = 0 implies that 𝑃 (𝑌 ≥ 0) = inf 𝑡 𝑀 (𝑡). Is the infimum attained in this case?
6. M(t) denotes the moment generating function of the random variable 𝑋. Show that
P(𝑋 > 𝑥) ≤ inf 𝑀 (𝑡)𝑒−𝑡𝑥 .
𝑡>0
Using this, give an estimate for P(𝑃 𝑂𝐼(𝜆) > 𝑥)!
1. We toss a fair coin. 𝑞𝑛 denotes the probability that in the sequence of first 𝑛 tosses
there are no three consecutive heads. (You cannot see HHH in the sequence). Calculate
the generating function of the sequence 𝑞𝑛 .
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