Question

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 ).

(b) Derive a formula for E(Y |X) and use it to compute E(X + Y ) as E(E(X + Y |X)).

Answer #1

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...

Toss five fair coins and let x be the number of tails
observed.
a. Calculate p(x) for the values x=2 and x=3.
b. Construct a probability histogram for p(x).
c. What is P(x=3 or x=4)?
Please show steps on how you get the answer!

Toss three fair coins and let x equal the number of tails
observed.
a. Identify the sample points associated with this experiment,
and assign a value of x to each sample point. Then list all the
possible values of x.
b. Calculate p(x) for the values x=1 and x=2.
c. Construct a probability histogram for p(x).
d. What is P(x=2 or x=3)?

Suppose you toss two fair coins, with four possible outcomes. x
= heads and y = tails
What is E(w) if w = 2x+3y?

Four quarters are tossed one-hundred times and the number of
tails that come up with each toss are recorded. The following
results were obtained: zero tails one tail two tails three tails
four tails 10 15 40 20 15 Test the null hypothesis with a 10%
significance level that the coins are fair.

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.)

Three dice are rolled and two fair coins are tossed. Let X be
the sum of the number of spots that show on the top faces of the
dice and the number of coins that land heads up. The expected value
of X is ____?

A fair six-sided die is rolled 10 independent times. Let X be
the number of ones and Y the number of twos.
(a) (3 pts) What is the joint pmf of X and Y?
(b) (3 pts) Find the conditional pmf of X, given Y = y.
(c) (3 pts) Given that X = 3, how is Y distributed
conditionally?
(d) (3 pts) Determine E(Y |X = 3).
(e) (3 pts) Compute E(X2 − 4XY + Y2).

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 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

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