Math tools 2015-6
Assignment # 3
Guy Kindler, Nitzan Guberman
Unbalanced Ramsey numbers
1. Recall that the Ramsey number R(s, t) is the smallest number n such that any
undirected graph on n vertices must contain either a clique with s vertices or an
independent set with t vertices. Here we consider the Ramsey numbers of the form
R(t, 2t). Prove that
2t
5
.
R(t, 2t) ≥
2
Hint: Show that in G(n, p), with an appropriate value of p, a random graph with
t
n = 52 2 vertices won’t contain a t-clique or a 2t-independent-set.
Rainy days
2. The weather in Kuala Lumpur behaves as follows:
• Each day is either warm and sunny or warm and rainy,
• the first day of 2016 is rainy or sunny, with probability 1/2 each,
• in every other day the weather is as in the day before it with probability 2/3,
or it is different with probability 1/3.
For i = 0, 1, 2, . . ., let Xi be 1 if the weather is sunny on the i-th day after the
beginning of 2016, and −1 otherwise. Also, let
(
X1
i=0
Yi =
Xi · Xi−1 i > 0.
(a) Compute the expectation of the number of rainy days in 2016.
(b) Show that the variance of the number of rainy days in the year is bounded by
200.
(c) Use the variance to compute an upper bound on the probability that the
number of rainy days throughout the year exceeds 250.
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(d) Suppose that on the first day of 2016, the weather is not random but is fixed
to be sunny instead (in celebration of the new year). Prove that even in that
1
case, lim Pr(Xi = 1) =
i→∞
2
Polya’s Urn
At time t = 0, the spoop-box has a white spoops and b black spoops. At every integral
time point thereafter, a uniformly random spoop is chosen out of all the spoops currently
in the box. The chosen spoop then divides into two adorable little baby spoops, whose
color is the same as their parent.
3. Let S be the spoop that is chosen at time t = 2015, right before it divides into two.
What is the probability that S is white? Write your answer as a function in a and
b.
4. Suppose a = b = 1, and note that in this case, at any integral number t, the number
of spoops just before the next division occurs is t + 2. Let At be the number white
of spoops at time t (this is a random variable). Show that at time t = 2015, At is
uniformly distributed over the set {1, 2, . . . , 2016}.
Concentration from moments
5. Let X be a real-valued random variable and suppose that for every p ≥ 1, E [|X|p ] ≤
t2
pp/2 . Show that for every t > 10, Pr [X > t] ≤ e− 2e .
Hint: get a different bound for each p and then optimize over p)
A Chernoff type bound for symmetric variables
6. Prove that for every x ∈ R,
exponential function).
ex +e−x
2
≤ ex
2 /2
(hint: use the Taylor series of the
7. Suppose {Xi }i=1,...,n is a collection of independent real valued random variables
such that
(
ai
w.p. 1/2
Xi =
−ai
w.p. 1/2
Give a direct proof (not using a known Chernoff-type bound) for the following: for
every λ > 0,
" n
#
λ2
X
− Pn
2
Pr
Xi > λ ≤ e 2 i=1 ai
i=1
(Hint: follow the scheme we used for proving the Chernoff type bound for Bernoulli
variables).
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