Binomial Distribution

Binomial Distribution - 4 Conditions

1. The experiment consists of 𝑛 identical trials.
2. Each trial results in one of the two outcomes, called success and failure.
3. The probability of success, denoted 𝑝, remains the same from trial to trial.
4. The 𝑛 trials are independent.

Recognizing Binomial Variables

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example	X = Binomial Variable?	why
in a game involving a standard deck of 52 playing cards, an individual randomly draws 7 cards without replacement. let X = the number of aces drawn	NO	trials are not independent
60% of a certain species of tomato live after transplanting from pot to garden. Eli transplants 16 of these tomato plants. Assume that the plants live independently of each other let X = the number of tomato plants that live	YES	fulfills all 4 conditions
in a game of luck, a turn consists of a player continuing to roll a pair of six-sided dice until they roll a double (2 same face values) let X = the number of rolls in one turn	NO	number of trials is not fixed

Probability Mass Function

𝐏(𝐗=𝑥|𝑛) defines the probability of obtaining exactly 𝑥 successes out of 𝑛 Bernoulli trials 𝑋₁, …, 𝑋_𝑛, where 𝑝 is the probability of success for a Bernoulli trial (i.e. 𝐏(𝑋_𝑖=1) = 𝑝)

• 𝐏(𝐗=𝑥;𝑛,𝑝) = [𝑛!/(𝑥!(𝑛-𝑥)!] * 𝑝^𝑥 * (1-𝑝)^(𝑛-𝑥)

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• 𝐏(𝐗=𝑥|𝑛) = [𝑛 choose 𝑥] * prob-of-𝑥-success * prob-of-(𝑛-𝑥)-failures
• 𝐏(𝐗=𝑥|𝑛) = [𝑛 choose 𝑥] * 𝐏(𝑋=1)^𝑥 * 𝐏(𝑋=0)^(𝑛-𝑥)
• 𝐏(𝐗=𝑥|𝑛) = [𝑛 choose 𝑥] * 𝑝^𝑥 * (1-𝑝)^(𝑛-𝑥)
• 𝐏(𝐗=𝑥|𝑛) = [𝑛!/(𝑥!(𝑛-𝑥)!] * 𝑝^𝑥 * (1-𝑝)^(𝑛-𝑥)

Cumulative Distribution Function

= 𝐼_𝑞(𝑛-𝑥, 1+𝑥)

where:

• 𝐼 is the Regularized Incomplete Beta Function
• 𝑞 = 1 - 𝑝

Expectation

𝐄[𝐗] = 𝑛𝑝

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The expectation of 1 Bernoulli Variable 𝑋with a probability of success 𝐏(𝑋=1) = 𝑝:

• 𝐄[𝑋] = 𝛴_𝑥∊𝑋[𝑥·𝐏(𝑋=𝑥)]
• 𝐄[𝑋] = (0)𝐏(𝑋_𝑖=0) + (1)𝐏(𝑋_𝑖=1)
• 𝐄[𝑋] = (0)(1−𝑝) + (1)(𝑝)
• 𝐄[𝑋] = 𝑝

The expectation of 𝑛 Bernoulli Variables 𝑋₁, …, 𝑋_𝑛 (i.e. Binomial Variable 𝐗):

• 𝐄[𝐗] = 𝐄[𝑋₁+ … + 𝑋_𝑛]
• 𝐄[𝐗] = 𝐄[𝑋₁] + … + 𝐄[𝑋_𝑛]
• 𝐄[𝐗] = 𝑝 + … + 𝑝
• 𝐄[𝐗] = 𝑛𝑝

Variance

𝑉𝑎𝑟(𝐗) = 𝑛𝑝(1-𝑝)

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variance of 1 Bernoulli Variable 𝑋_𝑖with probability of success 𝐏(𝑋_𝑖=1) = 𝑝

• 𝑉𝑎𝑟(𝑋_𝑖) = 𝐄[(𝑥 − 𝐄[𝑋_𝑖])²]
• 𝑉𝑎𝑟(𝑋_𝑖) = 𝛴_{𝑥∊𝑋𝑖}[(𝑥 − 𝐄(𝑋_𝑖))²𝐏(𝑋_𝑖=𝑥)]
• 𝑉𝑎𝑟(𝑋_𝑖) = 𝛴_{𝑥∊𝑋𝑖}[(𝑥−𝑝)²𝐏(𝑋_𝑖=𝑥)]
• 𝑉𝑎𝑟(𝑋_𝑖) = (0−𝑝)²𝐏(𝑋_𝑖=0) + (1−𝑝)²𝐏(𝑋_𝑖=1)
• 𝑉𝑎𝑟(𝑋_𝑖) = (0−𝑝)²(1−𝑝) + (1−𝑝)²𝑝
• 𝑉𝑎𝑟(𝑋_𝑖) = 𝑝²(1−𝑝) + (1−𝑝)²𝑝
• 𝑉𝑎𝑟(𝑋_𝑖) = (1−𝑝)(𝑝² + (1−𝑝)𝑝)
• 𝑉𝑎𝑟(𝑋_𝑖) = (1−𝑝)(𝑝² + 𝑝 − 𝑝²)
• 𝑉𝑎𝑟(𝑋_𝑖) = (1−𝑝)(𝑝)
• 𝑉𝑎𝑟(𝑋_𝑖) = 𝑝(1−𝑝)

variance of 𝑛 Bernoulli Variables 𝑋₁, …, 𝑋_𝑛 (i.e. Binomial Variable 𝐗):

• 𝑉𝑎𝑟(𝐗) = 𝑉𝑎𝑟(𝑋₁+ … + 𝑋_𝑛)
• 𝑉𝑎𝑟(𝐗) = 𝑉𝑎𝑟(𝑋₁) + … + 𝑉𝑎𝑟(𝑋_𝑛)
• 𝑉𝑎𝑟(𝐗) = 𝑝(1−𝑝) + … + 𝑝(1−𝑝)
• 𝑉𝑎𝑟(𝐗) = 𝑛𝑝(1-𝑝)

Moment-Generating Function$M_X(t)=[1-p+p{e}^t{]}^n$

See: Moment-Generating Function - Binomial Distribution

Subpages

• Geometric Distribution vs Binomial Distribution
• Negative Binomial Distribution vs Binomial Distribution
• Binomial Distribution vs Poisson Distribution
• Binomial Distribution Special Cases - (Poisson Distribution and Normal Distribution)