Question

1. Let X be the number of heads in 4 tosses of a fair coin.

(a) What is the probability distribution of X? Please show how probability is calculated.

(b) What are the mean and variance of X?

(c) Consider a game where you win $5 for every head but lose $3 for every tail that appears in 4 tosses of a fair coin. Let the variable Y denote the winnings from this game. Formulate the probability distribution of Y based on the probability distribution of X.

(d) What is the expected value of Y? Would you like to play this game? If so, why? If not, why not?

Answer #1

Let X be the number of heads in three tosses of a fair coin.
a. Find the probability distribution of Y = |X − 1|
b. Find the Expected Value of Y

A game consists of flipping a fair coin twice and counting the
number of heads that appear. The distribution for the number of
heads, X, is given by: P(X = 0) = 1/4; P(X =1) = 1/2; P(X = 2) =
1/4. A player receives $0 for no heads, $2 for 1 head, and $5 for 2
heads (there is no cost to play the game). Calculate the expected
amount of winnings ($).

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 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 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 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).

what is the probability of 2 heads in 2 fair coin tosses ?

Alan tosses a coin 20 times. Bob pays Alan $1 if the first toss
falls heads, $2 if the first toss falls tails and the second heads,
$4 if the first two tosses both fall tails and the third heads, $8
if the first three tosses fall tails and the fourth heads, etc. If
the game is to be fair, how much should Alan pay Bob for the right
to play the game?

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

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