Question

Using R, simulate tossing 4 coins as above, and compute the random variable X(the outcome of tossing a fair coin 4 times & X = num of heads - num of tails.). Estimate the probability mass function you computed by simulating 1000 times and averaging.

Answer #1

The outcome of tossing a fair coin 4 times will be 2^{4}
= 16 which are HHHH HHHT HHTH HTHH HHTT HTHT HTTT THHH TTHH THHT
TTHT HTTH THTH TTTH THTT TTTT.

Let X be the difference between number of heads and number of tails

So, the value of X can be (4-0), (3-1), (2-2), (3-1), (4-0) = 0, 2 ,4

The probability mass function

P(X=x)=

where x is the outcome and n is the number of coin tossed.

R code

#taking 0 =Tail 1=Head

n=4 #no of coin

FlipCoin = function(n) sample(0:1,n,rep=T)

e1=FlipCoin(16)

e1 # sample space

flips<-rbinom(1000, 4, .5)

pbinom(flips,n,.5) #probability mass function

OP:

> e1 [1] 0 0 0 1 1 1 1 1 1 0 0 1 1 0 1 1

pbinom(flips,n,.5) [1] 0.6875 0.3125 0.9375 0.6875 0.6875 0.3125 0.3125 0.3125 0.9375....

You select a coin at random: 2/3 of the coins are unfair, 1/3 of
the coins are fair. The fair coins are equally likely to flip heads
or tails. The unfair coins flip heads 3/4 of the times, and tails
1/4 of the times. You flip the selected coin and get heads or
tails. Find (1) the probability that the selected coin is fair
given the flip is heads, (2) the probability that the selected coin
is fair given the...

The random variable X is the number of tails obtained after
tossing 4 coins. A) List the all possible outcomes. How many
possible outcomes? ARe they equally likely outcomes? B) Using a
table, display the probability distribution of the number of tails.
C) Calculate the probability of one tail or two tails. D) Calculate
the probability of at least of one tail.

Consider an experiment of tossing two coins three times. Coin A
is fair but coin B is not with P(H)= 1/4 and P(T)= 3/4. Consider a
bivariate random variable (X,Y) where X denotes the number of heads
resulting from coin A and Y denotes the number of heads resulting
from coin B.
(a) Find the range of (X,Y)
(b) Find the joint probability mass function of (X,Y).
(c) Find P(X=Y), P(X>Y), P(X+Y<=4).
(d) Find the marginal distributions of X and...

An experiment consists of tossing a coin 6 times. Let X
be the random variable that is the number of heads in the outcome.
Find the mean and variance of X.

Use a random-number table to simulate the outcomes of tossing a
quarter 12 times beginning at row 1, block 4. Assume that the
quarter is balanced (i.e., fair) and an even digit is assigned to
the outcome heads (H) and an odd digit to the outcome tails
(T).
65321
85623
10204
50218
20321
22315
98532
91972
39800
45670
20510
10451
92012
59826
35456
79289
91483
29754
45652
98653
45863
36963
15326
78952
45678
10100
91251
37041
13712
14672

Conducting a Simulation
For example, say we want to simulate the probability of getting
“heads” exactly 4 times in 10 flips of a fair coin.
One way to generate a flip of the coin is to create a vector in
R with all of the possible outcomes and then randomly select one of
those outcomes. The sample function takes a vector of elements (in
this case heads or tails) and chooses a random sample of size
elements.
coin <- c("heads","tails")...

The probability with which a coin shows heads upon tossing is p.
The random variable X1 takes the values 1 and 0 if the outcome of
the "first toss is heads or tails respectively; another random
variable X2 is defined in the same way based on the second
toss.
(a) Is X1-X2 a sufficient statistic for p? Show the work.
(b) Is X1+X2 a sufficient estimator for p? Show the work.

In this problem, a fair coin is flipped three times. Assume that
a random variable X is defined to be 7 times the number of heads
plus 4 times the number of tails.
How many different values are possible for the random variable
X?

1. In this problem, a fair coin is flipped three times. Assume
that a random variable X is defined to be 7 times the number of
heads plus 4 times the number of tails.
How many different values are possible for the random variable
X?
2. Fill in the table below to complete the probability density
function. Be certain to list the values of X in ascending
order.
Value of X | Probability

Let X be the random variable representing the difference between
the number of headsand the number of tails obtained when a fair
coin is tossed 4 times.
a) What are the possible values of X?
b) Compute all the probability distribution of X?
c) Draw the cumulative distribution function F(x) of the random
variable X.

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