Question

**Question 1:** Let X = the number of “heads” after
5 flips of a fair coin. What is P(X > 1)? Please round your
answer to the nearest hundredth.

**Question 2:** The variance of X is 0.16. The
variance of Y is 0.01. What is the variance of W = X - 3Y? Assume
that X and Y are independent.

**Question 3:** Which of the following is not true
about the t distribution? Please choose the best answer:

a) The total area under the curve is 1

b) All of the statements are correct

c) The distribution is symmetric

d) It is a bell-shaped distribution

e) the mean is 0

Answer #1

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...

You flip a coin until getting heads. Let X be the number of coin
flips.
a. What is the probability that you flip the coin at least 8
times?
b. What is the probability that you flip the coin at least 8
times given that the first, third, and fifth flips were all
tails?
c. You flip three coins. Let X be the total number of heads. You
then roll X standard dice. Let Y be the sum of those...

Let X equal the number of flips of a fair coin that are required
to observe tails–heads on consecutive flips. d) Find E(X + 1)^2 (e)
Find Var(kX − k), where k is a constant

Question 3: You are
given a fair coin. You flip this coin twice; the two flips are
independent. For each heads, you win 3 dollars, whereas for each
tails, you lose 2 dollars. Consider the random variable
X = the amount of money that you
win.
– Use the definition of expected value
to determine E(X).
– Use the linearity of expectation to
determineE(X).
You flip this coin 99 times; these
flips are mutually independent. For each heads, you win...

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

Suppose Brian flips three fair coins, and let X be the number of
heads showing. Suppose Maria flips five fair coins, and let Y be
the number of heads showing. Let
Z = (X − Y) Compute P( Z = z)
.

a
fair coin is flipped 44 times. let X be the number if heads. what
normal distribution best approximates X?

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...

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 the probability of more than 30 heads in 50 flips of a fair
coin by using the normal approximation to the binomial
distribution.
a) Check the possibility to meet the requirements to use normal
approximation (show your calculation)
b) Find the normal parameters of the mean(Mu) and standard
deviation from the binomial distribution.
c) Apply normal approximation by using P(X>30.5) with
continuity correction and find the probability from the table of
standard normal distribution.

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