Question

Suppose we toss a fair coin n = 1 million times and write down the outcomes: it gives a Heads-and-Tails-sequence of length n. Then we call an integer i unique, if the i, i + 1, i + 2, i + 3, . . . , i + 18th elements of the sequence are all Heads. That is, we have a block of 19 consecutive Heads starting with the ith element of the sequence. Let Y denote the number of unique integers i. What is the expected value of Y?

Answer #1

Suppose we toss a fair coin three times. Consider the events
A: we toss three heads, B: we toss at least one
head, and C: we toss at least two tails.
P(A) = 12.5
P(B) = .875
P(C) = .50
What is P(A ∩ B), P(A ∩ C) and P(B ∩ C)?
If you can show steps, that'd be great. I'm not fully sure what
the difference between ∩ and ∪ is (sorry I can't make the ∪
bigger).

A fair coin is tossed for n times independently. (i) Suppose
that n = 3. Given the appearance of successive heads, what is the
conditional probability that successive tails never appear? (ii)
Let X denote the probability that successive heads never appear.
Find an explicit formula for X. (iii) Let Y denote the conditional
probability that successive heads appear, given no successive heads
are observed in the first n − 1 tosses. What is the limit of Y as n...

a) Suppose we toss a fair coin 20 times. What is the probability
of getting between 9 and 11 heads?
b) Suppose we toss a fair coin 200 times. What is the
probability of getting between 90 and 110 heads?

Toss a fair coin five times and record the results. Use T to
denote tails facing up and H to denote heads facing up.
(a) [5 points] How many different results can we get?
(b) [10 points] What is the probability that there is
only one T in the result? Round answer to 4
decimal .
(c) [10 points] What is the probability that at least
two T's in the result? Round answer to 4 decimal .

Suppose we toss a biased coin independently until a random time
N independent of the outcomes of the tosses. Where N takes values
1,2,3 with probability 0.3, 0.5, 0.2. Find E(X1 + ··· XN) where Xi
= 1 if head-on i th toss with probability 0.55 and zero otherwise,
(for i = 1, ··· , N).

Suppose we toss a fair coin twice. Let X = the number of heads,
and Y = the number of tails. X and Y are clearly not
independent.
a. Show that X and Y are not independent. (Hint: Consider the
events “X=2” and “Y=2”)
b. Show that E(XY) is not equal to E(X)E(Y). (You’ll need to
derive the pmf for XY in order to calculate E(XY). Write down the
sample space! Think about what the support of XY is and...

Q3. Suppose you toss n “fair” coins (i.e.,
heads probability = 1/21/2). For every coin that came up tails,
suppose you toss it one more time. Let X be the random variable
denoting the number of heads in the end.
What is the range of the variable X (give exact upper and lower
bounds)
What is the distribution of X? (Write down the name and give a
convincing explanation.)

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