1) A coin having probability p of coming up heads is continually flipped until both heads and tails have appeared. find (a) the expected number of flips (b) the probability that the last flip lands heads

2) Ten hunters are waiting for ducks to fly by. when a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. if each hunter independently hits his target with probability 0.6, compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poissonrandom variable with mean 6.

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1) Probability of getting head = p, probability of getting tail = q.
The the random variable follows Geometric distribution with pdf given as:

Expectationand variance of a random variable
Let X be a random variable with the following probability distribution: ...
[See the attached Question File.]

A game is played in which one of two players chooses a dice from a set of three die. After the first player chooses, the second player chooses a dice from the remaining two that are left. Then the two role their dice simultaneously. The player that turns up the highest number wins. In this game, however, although the dice are

A particle is constrained to lie along a one-dimensional line segment between 0 and a. The probability of finding the particle along this line segment is seen in attached file.
A. Find the proportionality constant, ie., normalize the function.
B. Calculate the expectation values of X and X^2 for n=1 and n=2.
C. Find t

4. An experiment is set to test the hypothesis that a given coin is unbiased. The decision rule is the following: Accept the hypothesis if the number of heads in a sample of 200 tosses is between 90 and 110 inclusive, otherwise reject the hypothesis.
a) Find the probability of accepting the hypothesis when it is correct.

A pollster wishes to obtain information on intended voting behavior in a two party system and samples a fixed number (n) of voters. Let X_1 ..., X_n denote the sequence of independent Bernoulli random variables representing voting intention, where E(X_l) = p, i = 1, ..., n
(a) Suppose the number of voters n is fixed, compute

We repeatedly throw a die, stopping when the value of the throw exceeds the value of the first throw. Compute the expectation value of the number of throws.

1. show that the commutator obeys:
[A,B] = -[B,A]
[A,B+C]=[A,B] + [A,C]
[A,BC]=[A,B]C+B[A,C]
[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0
Given the fundamental commutator relation between momentum and position [x,p] = ih
show that:
a. [x^n,p] = ihn*x^(n-1)
b. [x,p^n] = ihn*p^(n-1)
c. show that if f(x) can be expanded in polyno

Let x be a random variable with the following probability distribution:
Value x of X......P(X = x)
........-1...............0.05
.........0...............0.05
.........1...............0.60
.........2...............0.05
.........3...............0.15
.........4...............0.10
Find the expectation E(X) an

Markov chains. I am finding the continuous case a bit unintuitive. This problem concerns martingales and branching processes, but it is supposedly not hard. I just need a detailed solution to this problem, so that I can see what concepts are used to solve it.