In: Statistics and Probability

A fair coin is tossed four times. Let X denote the number of heads occurring and let Y denote the longest string of heads occurring. (i) determine the joint distribution of X and Y (ii) Find Cov(X,Y) and ρ(X,Y).

A fair coin is tossed r times. Let Y be the number of heads in
these r tosses. Assuming Y=y, we generate a Poisson random variable
X with mean y. Find the variance of X. (Answer should be based on
r).

An unbiased coin is tossed four times. Let the random variable X
denote the greatest number of successive heads occurring in the
four tosses (e.g. if HTHH occurs, then X = 2, but if TTHT occurs,
then X = 1).
Derive E(X) and Var(X).
(ii) The random variable Y is the number of heads occurring in
the four tosses. Find Cov(X,Y).

If a fair coin is tossed 25 times, the probability distribution
for the number of heads, X, is given below. Find the mean and the
standard deviation of the probability distribution using Excel
Enter the mean and round the standard deviation to two decimal
places.
x P(x)
0 0
1 0
2 0
3 0.0001
4 0.0004
5 0.0016
6 0.0053
7 0.0143
8 0.0322
9 0.0609
10 0.0974
11 0.1328
12 0.155
13 0.155
14 0.1328
15 0.0974
16 ...

A coin is tossed twice. Let Z denote the number of heads on the
first toss and W the total number of heads on the 2 tosses. If the
coin is unbalanced and a head has a 40% chance of occurring, find
the correlation between W and Z.

Flip a fair coin 100 times. Let X equal the number of heads in
the first 65 flips. Let Y equal the number of heads in the
remaining 35 flips.
(a) Find PX (x) and PY (y).
(b) In a couple of sentences, explain whether X and Y are or are
not independent?
(c) Find PX,Y (x, y).

An
honest coin is tossed n=3600 times. Let the random variable Y
denote the number of tails tossed. Use the 68-95-99.7 rule to
determine the chances of the outcomes. (A) Estimate the chances
that Y will fall somewhere between 1800 and 1860. (B) Estimate the
chances that Y will fall somewhere between 1860 and 1890.

Let X be the number of Heads when we toss a coin 3 times. Find
the probability distribution (that is, the probability function)
for X

HOMEWORK-You flip a coin FOUR times. Let H = Number of
Heads.
Calculate:
(a) P (H = 4) =
(b) P (H ≥ 1) =
(c) P (H < 4) =
(d) P (1 < H ≤ 4) =
HINT: GET YOUR SAMPLE SPACE.
Leave your answers EITHER as a simplified fraction ( e.g. 4/16 =
1/4 when simplified) OR a decimal rounded to FOUR
decimal places. Also do not forget to enter your leading zero when
entering decimals.
CAUTION: FOLLOW...

STAT 2332
#1. A coin is tossed 1000 times, it lands heads 516 heads, is
the coin fair?
(a) Set up null and alternative hypotheses (two tailed).
(b) Compute z and p.
(c) State your conclusion.
#2. A coin is tossed 10,000 times, it lands heads 5160 heads, is
the coin fair?
(a) Set up null and alternative hypotheses (two tailed).
(b) Compute z and p.
(c) State your conclusion.

1. Four coins are tossed 11 times. The number of heads is
counted and the following data is reported: 4, 3, 2, 0, 3, 3, 1, 2,
3, 2, 1.
Calculate the following sample statistics. Be sure to provide a
formula for your answer.
a. Sample mean?
b. Sample median?
c. Sample mode?
d. Sample range?
e. Sample variance?

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