Question

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 find those probabilities.)

Answer #1

Toss a fair coin for three times and let X be the number of
heads.
(a) (4 points) Write down the pmf of X. (hint: first list all
the possible values that X can take, then calculate the probability
for X taking each value.)
(b) (4 points) Write down the cdf of X.
(c) (2 points) What is the probability that at least 2 heads
show up?

(a) A fair coin is tossed five times. Let E be the event that an
odd number of tails occurs, and let F be the event that the first
toss is tails. Are E and F independent?
(b) A fair coin is tossed twice. Let E be the event that the
first toss is heads, let F be the event that the second toss is
tails, and let G be the event that the tosses result in exactly one
heads...

I toss 3 fair coins, and then re-toss all the ones that come up
tails. Let X denote the number of coins that come up heads on the
first toss, and let Y denote the number of re-tossed coins that
come up heads on the second toss. (Hence 0 ≤ X ≤ 3 and 0 ≤ Y ≤ 3 −
X.)
(a) Determine the joint pmf of X and Y , and use it to calculate
E(X + Y )....

PROBLEM 4. Toss a fair coin 5 times, and let X be the number of
Heads. Find P ( X=4 | X>= 4 ).

We toss a fair coin twice, define A ={the first toss is head}, B
={the second toss is tail}, and C={the two tosses have different
outcomes}. Show that events A, B, and C are pairwise independent
but not mutually independent.

A coin is tossed twice. Consider the following events.
A: Heads on the first toss.
B: Heads on the second toss.
C: The two tosses come out the same.
(a) Show that A, B, C are pairwise independent but not
independent.
(b) Show that C is independent of A and B but not of A ∩ B.

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).

Suppose you toss a fair coin three times. Which of the following
events are independent? Give mathematical justification for your
answer.
A=
{“heads on first toss”}; B=
{“an odd number of heads”}.
A=
{“no tails in the first two tosses”}; B=
{“no heads in the second and third
toss”}.

let x and y be
the random variables that count the number of heads and the number
of tails that come up when two fair coins are flipped. Show that X
and Y are not independent.

A fair coin is tossed three times. Let X be the number of heads
among the first two tosses and Y be the number of heads among the
last two tosses. What is the joint probability mass function of X
and Y? What are the marginal probability mass function of X and Y
i.e. p_X (x)and p_Y (y)? Find E(X) and E(Y). What is Cov(X,Y) What
is Corr (X,Y) Are X and Y independent? Explain. Find the
conditional probability mass...

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