Question

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.

Answer #1

We toss n coins and each one shows up heads with probability p,
independent of the other coin tosses. Each coin which shows up
heads is tossed again.
What is the probability mass function of the number of heads
obtained after the second round of coin tossing?

Let H (Heads) and T (Tails) denote the two outcomes of a random
experiment of tossing a fair coin. Suppose I toss the coin infinite
many times and divide the outcomes (which are infinite sequences of
Heads and Tails) into two types of events:
(a) the portion of H or T of is exactly one half
(e.g.HTHTHTHT... or HHTTHHTT...)
(b) the portion of H or T is not one half (i.e. the complement
of event (a). e.g. HTTHTTHTT...). What are...

it is known that the probability p of tossing heads on an
unbalanced coin is either 1/4 or 3/4. the coin is tossed twice and
a value for Y, the number of heads, is observed. for each possible
value of Y, which of the two values for p (1/4 or 3/4) maximizes
the probabilty the Y=y? depeding on the value of y actually
observed, what is the MLE of p?

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.

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.

An unfair coin is such that on any given toss, the probability
of getting heads is 0.6 and the probability of getting tails is
0.4. The coin is tossed 8 times. Let the random variable X be the
number of times heads is tossed.
1. Find P(X=5).
2. Find P(X≥3).
3. What is the expected value for this random variable?
E(X) =
4. What is the standard deviation for this random variable? (Give
your answer to 3 decimal places)
SD(X)...

You have a coin that you suspect is not fair (i.e., the
probability of tossing a head (PH) is not equal to the probability
of tossing a tail (PT) or stated in another way: PH ≠ 0.5). To
test yoursuspicion, you record the results of 25 tosses of the
coin. The 25 tosses result in 17 heads and 8 tails.
a) Use the results of the 25 tosses in SPSS to construct a 98%
confidence interval around the probability of...

Suppose I have two biased coins: coin #1, which lands heads with
probability 0.9999, and coin #2, which lands heads with probability
0.1. I conduct an experiment as follows. First I toss a fair coin
to decide which biased coin I pick (say, if it lands heads, I pick
coin #1, and otherwise I pick coin #2) and then I toss the biased
coin twice. Let A be the event that the biased coin #1 is chosen,
B1 the event...

An ordinary (fair) coin is tossed 3 times. Outcomes are thus
triples of "heads" (h) and "tails" (t) which we write hth, ttt,
etc. For each outcome, let R be the random variable counting the
number of heads in each outcome. For example, if the outcome is
ttt, then =Rttt0. Suppose that the random variable X is defined in
terms of R as follows: =X−R4. The values of X are thus:
Outcome
tth
hth
htt
tht
thh
ttt
hht
hhh...

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?

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