Question

A coin with P[Heads]= p and P[Tails]= 1p is tossed repeatedly (the tosses are independent). Deﬁne (X = number of the toss on which the ﬁrst H appears, Y = number of the toss on which the second H appears. Clearly 1X<Y. (i) Are X and Y independent? Why or why not? (ii) What is the probability distribution of X? (iii) Find the probability distribution of Y . (iv) Let Z = Y X. Find the joint probability mass function

Answer #1

A coin with P[Heads]= p and P[Tails]= 1p is tossed repeatedly
(the tosses are independent). Deﬁne (X = number of the toss on
which the ﬁrst H appears, Y = number of the toss on which the
second H appears. Clearly 1X<Y. (i) Are X and Y independent?
Why or why not? (ii) What is the probability distribution of X?
(iii) Find the probability distribution of Y . (iv) Let Z = Y X.
Find the joint probability mass function

A biased coin is tossed repeatedly. The probability of getting
head in any particular toss is 0.3.Assuming that the tosses are
independent, find the probability that 3rd head appears exactly at
the 10th toss.

A coin is tossed repeatedly until heads has occurred twice or
tails has occurred twice, whichever comes first. Let X be the
number of times the coin is tossed.
Find: a. E(X). b. Var(X).
The answers are 2.5 and 0.25

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A coin is tossed with P(Heads) = p
a) What is the expected number of tosses required to get n
heads?
b) Determine the variance of the number of tosses needed to get
the first head.
c) Determine the variance of the number of tosses needed to get
n heads.

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...

(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...

A fair coin has been tossed four times. Let X be the number of
heads minus the number of tails (out of four tosses). Find the
probability mass function of X. Sketch the graph of the probability
mass function and the distribution function, Find E[X] and
Var(X).

Suppose a coin is tossed three times and let X be a random
variable recording the number of times heads appears in each set of
three tosses. (i) Write down the range of X. (ii) Determine the
probability distribution of X. (iii) Determine the cumulative
probability distribution of X. (iv) Calculate the expectation and
variance of X.

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