Question

Problem 1: Relations among Useful Discrete Probability
Distributions. A **Bernoulli experiment** consists of
only one trial with two outcomes (success/failure) with probability
of success p. The Bernoulli distribution

is

**P (X = k) =
p**^{k}**q**^{1-k}**,
k=0,1**

The sum of n independent Bernoulli trials forms a binomial
experiment with parameters n and p. The binomial probability
distribution provides a simple, easy-to-compute approximation with
reasonable accuracy to hypergeometric distribution with parameters
N, M and n when n/N is less than or equal to 0.10. In this case, we
can approximate the hypergeometric probabilities by a binomial
distribution with parameters n and p = M/N. Further, the
**Poisson distribution** with mean μ = np gives an
accurate approximation to binomial probabilities when n is large
and p is small.

This is my third time posting this question the first two times
the same person answered it I would like a different person to
answer this time so **if you have already answered this
question please do not answer it again.**

**1. Suppose in a region in Saskatchewan, among a group of
20 adults with cancer, seven were physically abused during their
childhood. A random sample of five adult persons is taken from this
group. Assume that sampling occurs without replacement, and the
random variable X represents the number of adults in the sample who
were abused during their childhood period.**

**(a) Write the formula for p(x), the probability
distribution of X. How this distribution is called?**

**(b) Using the adequate formulas, find the mean and
variance of X?**

**(c) Find the probabilities of all the possible values of
X. Plot the histogram of X and try the locate the approximative
value of the mean μ.**

**(d) What is the probability that at least one person was
abused during childhood?**

**Now suppose another survey in British Columbia reveals
that among 180 adults with cancer, only 80 adults were abused in
their childhood. Suppose again that a random sample of five adult
persons is taken from this group without replacement and let denote
by Y the random variable which represents the number of adults
abused during their childhood period in the sample.**

**(e) Find the probabilities of all the possible values of
Y and plot the histogram of Y. How do you compare this histogram
with the histogram of X.**

**(f) Find the probabilities of all the possible values of
Y using the formula for the binomial distribution with p =
80/180**

**as an approximation. Plot the histogram and compare it
with the histogram obtained using the hypergeometric
formula.**

**(g) Is the precedent approximation close enough? Why or
why not?**

**(h) Calculate the mean and variance using both binomial
and hypergeometric distributions, respectively. Provide a
comparison and summarize your findings**

Answer #1

(1) Consider X that follows the Bernoulli distribution with
success probability 1/4, that is, P(X = 1) = 1/4 and P(X = 0) =
3/4. Find the probability mass function of Y , when Y = X4 . Find
the second moment of Y . (2) If X ∼ binomial(10, 1/2), then use the
binomial probability table (Table A.1 in the textbook) to find out
the following probabilities: P(X = 5), P(2.9 ≤ X ≤ 4.9) (3) A deck
of...

1.) A binomial experiment consists of 19 trials. The
probability of success on trial 12 is 0.54. What is the probability
of success on trial 16?
0.54
0.15
0.39
0.88
0.5
0.11
2. Assume that 12 jurors are randomly selected from a
population in which 86% of the people are Asian-Americans. Refer to
the probability distribution table below and find the indicated
probabilities.
xx
P(x)P(x)
0
0+
1
0+
2
0+
3
0+
4
0+
5
0.0004
6
0.0028
7...

1. Given a discrete random variable, X , where the
discrete probability distribution for X is given on right,
calculate E(X)
X
P(X)
0
0.1
1
0.1
2
0.1
3
0.4
4
0.1
5
0.2
2. Given a discrete random variable, X , where the
discrete probability distribution for X is given on right,
calculate the variance of X
X
P(X)
0
0.1
1
0.1
2
0.1
3
0.4
4
0.1
5
0.2
3. Given a discrete random variable, X...

A biased coin (one that is not evenly balanced) is tossed 6 times.
The probability of Heads on any toss is
0.3. Let X denote the number of Heads that come up.
1. Does this experiment meet the requirements to be considered a
Bernoulli Trial? Explain why or why
not.
2. If we call Heads a success, what would be the parameters of the
binomial distribution of X?
(Translation: find the values of n and p)
3. What is the...

Provide an example of a probability distribution of discrete
random variable, Y, that takes any 4 different integer values
between 1 and 20 inclusive; and present the values of Y and their
corresponding (non-zero) probabilities in a probability
distribution table.
Calculate:
a) E(Y)
b) E(Y2 ) and
c) var(Y).
d) Give examples of values of ? and ? , both non-zero, for a
binomial random variable X. Use either the binomial probability
formula or the binomial probability cumulative distribution tables...

For each of the random quantities X,Y, and Z, defined below
(a) Plot the probability mass function PMS (in the discrete
case) , or the probability density function PDF (in the continuous
case)
(b) Calculate and plot the cumulative distribution function
CDF
(c) Calculate the mean and variance, and the moment function
m(n), and plot the latter.
The random quantities are as follows:
X is a discrete r.q. taking values k=0,1,2,3,... with probabilities
p(1-p)^k, where p is a parameter with...

True or False:
10. The probability of an event is a value which must be greater
than 0 and less than 1.
11. If events A and B are mutually exclusive, then P(A|B) is always
equal to zero.
12. Mutually exclusive events cannot be independent.
13. A classical probability measure is a probability assessment
that is based on relative frequency.
14. The probability of an event is the product of the probabilities
of the sample space outcomes that correspond to...

1. Compute the mean and variance of the following probability
distribution. (Round your answers to 2 decimal
places.)
x
P(x)
4
0.10
7
0.25
10
0.30
13
0.35
2. Given a binomial distribution with n = 6 and π=
.25. Determine the probabilities of the following events using the
binomial formula. (Round your answers to 4 decimal
places.)
x = 2
x = 3
3. A probability distribution is a listing of all the outcomes
of an experiment and the...

Question 1 of 3 Which of the following is true about the
random variables X, Y, and Z?
X is binomial with n = 20 and p = .22.
Y is binomial with n = 40 and p = .32.
Z is not binomial. All of the above are true.
Only (A) and (B) are true.
Question 2 of 3 What is the probability that exactly 2
of the 20 older adults prefer organic? (Note: Some answers are
rounded.)
.105...

Question 1
Refer to the probability function given in the following table
for a
random variable X that takes on the values 1,2,3 and 4
X 1 2 3 4
P(X=x) 0.4 0.3 0.2 0.1
a) Verify that the above table meet the conditions
for being a discrete probability
distribution
b) Find P(X<2)
c) Find P(X=1 and X=2)
d) Graph P(X=x)
e) Calculate the mean of the random variable
X
f) Calculate the standard deviation of the random
variable X...

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