Binomial Distribution Analysis

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Questions and Answers

What is the probability mass function for a binomial distribution with parameters n and p?

$P(X = x) = (n - x) p^x q^{n-x}$
$P(X = x) = (n choose x) p^{n-x} q^x$
$P(X = x) = p^x + q^{n-x}$
$P(X = x) = (n choose x) p^x q^{n-x}$ (correct)

In the binomial distribution X ~ B(n, p), what do the parameters n and p represent?

The number of trials and the probability of success (correct)
The variance and the mean of the distribution
The average success rate and the minimum success needed
The total number of successes and the total number of failures

Which statement about the binomial distribution is FALSE?

It is a discrete distribution.
The outcomes are mutually exclusive.
The formula derives from the binomial expansion.
It can take on negative values. (correct)

What is the expected frequency of successes in a binomial distribution when N trials are conducted?

$N (n choose x) p^x q^{n-x}$ (B) Signup and view all the answers

The sum of probabilities over all possible outcomes in a binomial distribution equals what value?

1 (C) Signup and view all the answers

When n trials are performed, which of the following expressions is NOT a term in the binomial expansion?

$q^{n-1} p^{n}$ (B) Signup and view all the answers

What type of random variable follows a binomial distribution?

Only discrete integer values (A) Signup and view all the answers

Which of these conditions is necessary for a binomial distribution to be applicable?

Each trial has two exhaustive and mutually exclusive outcomes. (B) Signup and view all the answers

What is the mode of a binomial distribution when the number of trials n is odd?

The modes are at X = (n + 1) and X = (n - 1) (D) Signup and view all the answers

What is the moment generating function of a binomially distributed random variable X ~ B(n, p)?

M_x(t) = (q + p e^t)^n (C) Signup and view all the answers

Which of the following is NOT a characteristic of the second moment (μ2) of a binomial distribution?

μ2 = n^2p^2 (C) Signup and view all the answers

In the moment generating function of a binomial distribution, what term represents the contribution of the mean?

E[e^{tX}] (B) Signup and view all the answers

How is the third moment (μ3) of a binomial distribution characterized?

μ3 = npq(q - p) (B) Signup and view all the answers

What represents the mean deviation about the mean of the binomial distribution?

$n = ext{sum of deviations from mean}$ (B) Signup and view all the answers

For what value of $r$ is the factorial moment of the binomial distribution defined?

r > 0 (A) Signup and view all the answers

Which expression represents the first factorial moment $ ext{E}(X_{(1)})$ of the Binomial distribution?

$np$ (D) Signup and view all the answers

What is the relationship defined for $ ext{E}(X_{(3)})$ with respect to the previous moments?

$ ext{E}(X_{(3)}) = n(n-1)(n-2)p^3 - 3n(n-1)p^2 + np$ (C) Signup and view all the answers

What simplifies to $-2npq(1+p)$ in the context of moments for the Binomial distribution?

$ ext{E}(X_{(3)})$ expression simplification$ (D) Signup and view all the answers

In the equation $p rac{d}{dp} (q + p)^{n}$, what is being represented?

The first derivative of the generating function (D) Signup and view all the answers

What does $rac{d}{dp}(q + p)^{n}$ equal to for any integer transformation of $x$?

$ ext{Factorial moment for } r=k$ (D) Signup and view all the answers

The expression $rac{x!}{(x-r)!}$ indicates what in terms of factorial moments?

The contribution of $r$th order to the moment (B) Signup and view all the answers

What characteristics define Bernoulli trials?

The probability of success remains constant for each trial. (B) Signup and view all the answers

For the probability of getting at least seven heads when tossing ten coins, which components are included in the calculation?

p(7), p(8), p(9), p(10) (B) Signup and view all the answers

In a binomial experiment with five games played, if the probability of A winning each game is $rac{3}{5}$, what is the probability of A winning exactly two games?

0.0768 (C) Signup and view all the answers

What is the significance of binomial distribution in probability theory?

It helps derive many other probability distributions. (A) Signup and view all the answers

If A and B have winning chances in the ratio of 3:2, what is B's probability of winning?

$rac{2}{5}$ (C) Signup and view all the answers

In the case of throwing ten coins simultaneously, the probability of not getting a head is defined as what?

$rac{1}{2}$ (A) Signup and view all the answers

What does the binomial probability formula output when n = 10 and p = $rac{1}{2}$?

It computes the likelihood of successes for discrete trials. (C) Signup and view all the answers

If the probabilities of 1 and 2 successes in a binomial experiment of 5 trials are 0.4096 and 0.2048, respectively, what does this imply about the distribution?

The parameter p can be calculated from given probabilities. (A) Signup and view all the answers

What does the mode of a binomial distribution represent?

The value of x for which p(x) is maximum (C) Signup and view all the answers

In the case where (n + 1)p is an integer, what are the modal values of a binomial distribution?

m and m - 1 (D) Signup and view all the answers

If the mean of a binomial distribution is 4 and the variance is 3, what is the value of n?

16 (C) Signup and view all the answers

What inequality holds for p(x) when (n + 1)p is not an integer for values of x from 0 to m?

p(x) > p(x-1) (B) Signup and view all the answers

When $(n + 1)p$ is represented as $m+f$, what does 'f' signify?

A fractional part such that 0 < f < 1 (D) Signup and view all the answers

For p = 0.5 and n odd, where does the maximum probability of the binomial distribution occur?

At both (n+1) and (n-1) (B) Signup and view all the answers

What relationship confirms the probabilistic behavior of p(x) based on the index x?

p(x) follows a specific increasing and decreasing sequence (C) Signup and view all the answers

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Study Notes

Mode of Binomial Distribution

The mode is the value of x where the probability p(x) is maximum
There are two cases:
- Case 1: When (n+1)p is not an integer
  - The mode is the integral part of (n+1)p
- Case 2: When (n+1)p is an integer
  - The distribution is bimodal
  - The two modes are (n+1)p and (n+1)p - 1

Example 8.15

The given binomial distribution has parameters n=16 and p=1/4

Example 8.16

When p = 0.5, if n is even the binomial distribution peaks at x=n/2
When p = 0.5, if n is odd the binomial distribution peaks at x=(n+1)/2 and x=(n-1)/2

Factorial Moments of Binomial Distribution

The rth factorial moment of the binomial distribution is:
- n(r)p(r)

Mean Deviation about Mean of Binomial Distribution

Mean deviation is calculated by taking the sum of the absolute differences between each value and the mean, weighted by their probabilities.
The mean deviation is 2 * npq * (n-1) / (μ-1) * p^(μ-2) * q^(n-μ)

Binomial Probability Distribution

This occurs when there are n independent trials, each with a probability of success p.
The probability of x successes in n trials is given by:
- p(x) = (n choose x) * p^x * q^(n-x)
The two parameters n and p determine the distribution.
This distribution is discrete as it only takes on integer values between 0 and n.

Remarks on Binomial Distribution

Probabilities of all possible outcomes of a binomial distribution add up to 1.
If the experiment is repeated N times, the frequencies of each outcome are given by:
- f(x) = N * (n choose x) * p^x * q^(n-x)
The binomial distribution applies to situations with these properties:
- There are only two possible outcomes for each trial (success or failure).
- The number of trials is fixed and finite.
Trials are independent.
The probability of success is constant for each trial.
Examples include tossing coins or drawing cards with replacement.
Many other probability distributions are related to the binomial distribution.

Example 8.1

The probability of getting at least seven heads in ten coin tosses is 11/64.

Example 8.2

The probability of A winning at least three games out of five is 3/5.

Example 8.9

The parameter p of the binomial distribution is 0.8.

Moment Generating Function of Binomial Distribution

The moment generating function of the binomial distribution is:
- M_x(t)= (q + pe^t)^n
The moment generating function about the Mean of the binomial distribution is:
- (qe^(-pt) + pet)^n

Example 8.17

The distribution of X-np follows a shifted binomial distribution.

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