Question

Let W be a random variable giving the number of heads minus the number of tails in three independent tosses of an unfair coin where p = P(H) = 1 3 , and q = P(T) = 2 3 . (a) List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value of W. (b) Find P(−1 ≤ W < 1). (c) Draw a graph of the probability density function f(t) of W, and the cumulative distribution function F(t). (d) Compute µW = E(W) and σ 2 W .

Answer #1

Let W be a random variable giving the number of
tails
minus the number of
heads
in three tosses of a coin. Assuming that a
tail
is
half
as likely to occur, find the probability distribution of the
random variable W.
Complete the following probability distribution of W.

A fair coin has been tossed four times. Let X be the number of
heads minus the number of tails (out of four tosses). Find the
probability mass function of X. Sketch the graph of the probability
mass function and the distribution function, Find E[X] and
Var(X).

let X be the
random variable that equals the number of tails minus the number of
heads when n biased coins are flipped (probability for head is
2/3). What is the expected value of X? what is the variance of
X?

let x be the variable that results from determining the number
that represents two and a half times the number of shields minus
one and a half times the number of crowns in three launches of a
coin. List the elements for the 3 coin tosses and assign a value x
from X to each sample point. determine the probability
distribution, the mean of the distribution, the aculated
distribution and the variance

Let p denote the probability that a particular coin will show
heads when randomly tossed. It is not necessarily true that the
coin is a “fair” coin wherein p=1/2. Find the a posteriori
probability density function f(p|TN ) where TN is the observed
number of heads n observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this interval. Make
some plots of the a posteriori density.

a biased coin tossed four times P(T)=2/3 x is number
of tails observed
construct the table of probabulity function f(x) and cumulative
distributive function F(x)
and the probability that at least on tail is observed ie
P(X>1)

Assume p represents the probability that a particular
coin will show heads when randomly tossed. Don't assume its true
that the coin is a “fair” coin wherein p=1/2. Determine
the a posteriori probability density function
f(p|TN) where
TN is the observed number of heads n
observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this
interval. Create some plots of the a posteriori
density.

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

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

Deriving fair coin flips from biased coins: From coins with
uneven heads/tails probabilities construct an experiment for which
there are two disjoint events, with equal probabilities, that we
call "heads" and "tails".
a. given c1 and c2, where c1 lands heads up with probability 2/3
and c2 lands heads up with probability 1/4, construct a "fair coin
flip" experiment.
b. given one coin with unknown probability p of landing heads
up, where 0 < p < 1, construct a "fair...

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