Question

Bernoulli Trials and Binomial Distribution of probability.

Answer 1

Bernoulli trials:

The outcome of any trial is independent of the outcome of any other trial. In each of such trials, the probability of success or failure, remains constant. Such independent trials which have only two outcomes usually referred as 'success' or 'failure' are called Bernoulli trials.

Trials of a random experiment are called Bernoulli trials, if they satisfy the following conditions:

• There should be a finite number of trials.
• The trials should be independent.
• Each trials has exactly two outcomes: success or failure.
• The probability of success remains the same in each trial.

Binomial distribution:
The probability distribution of number of successes in an experiment consistiing of n Bernoulli trials may be obtained by distribution of binomial expansion of (q + p)". Hence, this distribution of number of successes X can be written as:

The above probability distribution is known as binomial distribution with parameters n and p, because for given values of n and p, we can find the complete probability distribution.

The probability of x successes P(X = x) is also denoted by P(x) and is given by

P(x) = ⁿC_x q^n-x p^x

x =0, 1,...., n{q = 1 - p)

This P(x) is called the probability function of the binomial distribution. A binomial distribution with n-Bernoulli trials and probability'’ of success in each trial as p, is denoted by B (n, p).

→ The conditional probability of an event A, given the occurrence of the given event B is given by:

→ If A and B be two events of sample space S and F be any other events such that P(F) ≠ 0, then

Specially, if A and B are disjoint events, them

P(A ∪ B)/F) = P(A/F) + P(B/F)

→ Multiplication Theorem on Probability

P(A ∩ B) = P(A)P \((\frac{B}{A})\)

P(A) ≠ 0 or P(A ≠ B)

= P(B)P \((\frac{A}{B})\); P(B) ≠ 0.

→ If A and B be two independent events, then

P \((\frac{A}{B})\) = P(A),P(A) ≠ 0;

 P \((\frac{A}{B})\) = P(A),P(B) ≠ 0;

and P(A ∩ B) = P(A)P(B).

→ Theorem of Total Probability: Let {A₁ A₂,....... A_n} be a partition of a sample space and suppose that each of A₁ A₂,....... A_n has non-zero probability. Let E be any event associated with S, then

→ Bayes' Theorem : If A₁ A₂,...... A_n are events which constitute a partition of sample space, S, i.e., A₁ A₂,...... A_n are pairwise disjoint and A₁ ∪ A₂ ∪ ...... ∪ A_n = S and E be any event with non-zero probability, then

→ A random variable is a real valued function whose domain is the sample space of a random experiment.

→ The probability distribution of a random variable X is the system of numbers:

→ Let X be a random variable whose possible values x₁ x₂, ........ x_n occur with probabilities p₁, p₂, ....... p_n respectively.

(a) The mean of X, denoted by p, is the number \(​​\sum^n_{i=1}\) x_ip_i

The mean of a random variable X is also called the expectation of X, denoted by E(X).

(b) Variance of X = var(X) = σ_x² = E(X - µ)² = \(​​\sum^n_{i=1}\) (x_i - µ)² p_i

(c) var(X) =E(X²) - {E(X)}²

The non-negative number

→ Trials of-a random experiment are called Bernoulli trials, if they satisfy the following conditions :

• There should be a finite number of trials.
• The trials should be independent.
• Each trial has exactly two outcomes success or failure.
• The probability of success remains the same in each trial.

For Binomial distribution B{n, p), P(X = x) = ⁿC_x q^n-xp^x, x = 0,1,... n (where q = 1 - p)