Test: Binomial Distribution - Civil Engineering (CE) MCQ

Concept:

Binomial distribution: If a random variable X has binomial distribution as B (n, p) with n and p as parameters, then the probability of random variable is given as:

Where, n is number of observations, p is the probability of success & (1 - p) is probability of failure.

Properties:

• Mean of the distribution (μ_X) = n × p
• The variance (σ²_x) = n × p × (1 - p)
• Standard deviation (σx) = 

Calculation:

Given:

mean of BD = np = 15

And variance of BD = npq = 10

⇒ np(1 – p) = 10 (∵ p + q = 1)

Binomial distribution:

Let p is the probability that an event will happen in a single trail (called the probability of success) and

q = 1 – p is the probability that an event will fail to happen (probability of failure)

The probability that the event will happen exactly r times in n trails (i.e. x successes and n – r failures will occur) is given by the probability function

where the random variable X denotes the number of successes in n trials and r = 0, 1, 2, … n

For Binomial distribution,

Mean = μ = np

Variance = σ² = npq

Standard deviation = σ = √(npq)

The expected value is sometimes known as the first moment of a probability distribution. The expected value is comparable to the mean of a population or sample.

Therefore, the first moment about the origin of the binomial distribution is,

Mean = μ = np

Concept:

Binomial distribution

where

p = Probability of success in one trial

q = Probability of failure in one trial = 1 – p

n = Total number of independent trials

k = Discrete random variable

Calculation:

n = 3

Using binomial distribution

Probability of getting red on the top face at least twice is

P(x ≥ 2) = P(x = 2) + P(x = 3)

We know Binomial distribution is:

where p + q = 1

p is the probability of getting success and q is the probability of failure

• Mean of Binomial distribution is np
• Variance is npq.
• Standard deviation is given by the square root of the variance, as:
  

Concept:

Binomial distribution: If a random variable X has binomial distribution as B (n, p) with n and p as parameters, then the probability of random variable is given as:

Where, n is number of observations or trials , p is the probability of success & (1 - p) or q is probability of failure.

Properties:

• Mean of the distribution (μX) = n × p
• The variance (σ²_x) = n × p × (1 - p)
• Standard deviation (σ_x) = √{np(1 - p)}
• Expectation, E(X) = np
• 1 - p = q

Calculation:

Given,

Expectation E(X) = 7 = np         .........(1)

Variance = 6 = npq                    .........(2)

From equation (1) & (2)

q = 6 / 7

∴ p = 1 / 7

Binomial Distribution:

A binomial distribution is a common probability distribution that occurs in practice. It arises in the following situation:

• There are n independent trials.
• Each trial results in a "success" or "failure"
• The probability of success in each and every trial is equal to 'p'.

If the random variable X counts the number of successes in the n trials, then X has a binomial distribution with parameters n and p.

X ~ Bin (n, p).

Properties of Binomial distribution:

If X ~ Bin (n, p), then the probability mass function of the binomial distribution is

f (x) = P (X =x) = nCr p^x(1 - p)^{n - x}

for x = 0, 1, 2, 3,...,n

Mean E (X) = μ = np.

Variance (σ²) = np(1 - p).

Note:

Theorem:

Let X₁, X₂, ..., X_m be independent random variables such that X_i has a BIn (n_i, p) distribution, for i = 1, 2, ..., m. Let

Bernoulli Distribution:

• A Bernoulli experiment/trial has only two possible outcomes, e.g. success/failure, heads/tails, female/male, defective/non-defective, etc.
• The outcomes are typically coded as  0 (failure) or 1 (success).

X ~ Bern (p)

P (X = 1) = 1, P (X = 0) = 1 - p, 0≤p≤1

Properties:

• The probability mass function is p(x) = p^x (1 - p)^{1 - x} for x = 0, 1.
• The mean is E (X) = μ = (1 × p) + 0 × (1 - p) = p
• Since E (X²) = (1² × p) + 0² × (1 - p) = p,
• σ² = var (X) = E (X²) - μ² = p - p² = p (1 - p). 

​Note:

The Bernoulli distribution is a special case of binomial distribution with n = 1.

Exponential Distribution:

The probability density function of the exponential distribution is,

Normal Distribution:

The probability density function of normal distribution is given by,

Concept:

Since both the positive and negative values are likely to occur, the negative values are binomially distributed

Binomial distribution:

 If a random variable X has binomial distribution as B (n, p) with n and p as parameters, then the probability of random variable is given as:

Where,

n is the number of observations, k is the number of success

p is the probability of success & (1 - p) is the probability of failure.

Calculation:

Given:

n = 5, p = 1/2

Let X represents number of negative values in 5 trials

At most 1 negative value means, k can be 0 and 1

P(At most one negative value) = P(X≤1)

P(X ≤ 1) = P(X = 0) + P(X = 1)

Important Points

• At most mean maximum. This term usually used when there are multiple trials like tossing a coin, throwing a dice
• For example in tossing two coins
• At most two head means it can be two head or one head or no head

Concept:

The probability that exactly 2 of the chosen items are defective is given as,

By Binomial distribution,

where, p = probability of success, q = probability of failure

Calculation:

Given:

n = 10, x = 2, p = 0.1, q = 0.9

Therefore, P(exactly 2 of the chosen items are defective) = ¹⁰C₂ × 0.1² × 0.9^{10 - 2} ⇒ 45 × 0.01 × 0.43 = 0.1937

Clearly, x is a binomial variate with parameters n and p = 1/2 such that

Now, P (x = 4), P(x = 5) and P(x = 6) are in AP.

If a, b, c are in AP then 2b = a+c

∴ 2P (x = 5) = P (x = 4) + P (x = 6)

⇒ n² – 21n + 98 = 0 ⇒ (n – 7) (n – 14) = 0

∴ n = 7 or 14 

Data

p = probability of success event = 0.4

q = probability of failure event = 0.6

n = Number of trials

Formulae: 

Standard Deviation = 

Variance in Binomial Distribution: npq 

Calculation: