UNIT-IV: Common Discrete and Continuous Probability Distributions

Binomial Distribution

The Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials.

Definition

A random variable X has a Binomial distribution if it represents the number of successes in n independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success (p) is constant for every trial.

Conditions for Binomial Distribution

• The experiment consists of a fixed number of trials, denoted by n.
• Each trial has only two possible outcomes: Success (S) or Failure (F).
• The trials are independent of each other.
• The probability of success, p, remains constant for each trial. Consequently, the probability of failure is q = 1 - p.

Parameters

• n: Number of trials.
• p: Probability of success on a single trial.

Probability Mass Function (PMF)

The probability of getting exactly k successes in n trials is given by:

P(X = k) = C(n, k) × p^k × (1-p)^n-k

Where:

• X is the random variable representing the number of successes.
• k is the specific number of successes we are interested in (k = 0, 1, 2, ..., n).
• C(n, k) (also written as ⁿC_k or (ⁿ_k)) is the binomial coefficient, calculated as n! / (k! × (n-k)!).
• p is the probability of success.
• (1-p) or q is the probability of failure.

Mean and Variance

Mean (Expected Value), E(X) = n × p
Variance, Var(X) = n × p × (1-p)

Applications

• Counting the number of defective items in a batch of manufactured products.
• Number of heads when flipping a coin n times.
• Number of patients recovering from a specific illness with a known recovery rate.
• Number of correct answers on a multiple-choice test by guessing.

Example

Suppose a biased coin has a probability of landing heads as 0.6. If you flip the coin 5 times, what is the probability of getting exactly 3 heads?

• n = 5 (number of trials)
• p = 0.6 (probability of success - getting a head)
• k = 3 (number of successes)

P(X = 3) = C(5, 3) × (0.6)³ × (1-0.6)^5-3

P(X = 3) = (5! / (3! × 2!)) × (0.6)³ × (0.4)²

P(X = 3) = 10 × 0.216 × 0.16

P(X = 3) = 0.3456

So, there is a 34.56% chance of getting exactly 3 heads.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, if these events occur with a known constant mean rate and independently of the time since the last event.

Definition

A random variable X follows a Poisson distribution if it represents the number of events occurring in a fixed interval of time or space, provided these events occur independently and at a constant average rate. It is often used for rare events.

Conditions for Poisson Distribution

• Events occur independently.
• The average rate of events (λ) is constant over the interval.
• The probability of two or more events occurring in a very short sub-interval is negligible.

Parameter

• λ (lambda): The average number of events in the given interval (mean rate).

Probability Mass Function (PMF)

The probability of observing exactly k events in the given interval is:

P(X = k) = (λ^k × e^-λ) / k!

Where:

• X is the random variable representing the number of events.
• k is the actual number of events we are interested in (k = 0, 1, 2, ...).
• λ is the average rate of event occurrence.
• e is Euler's number (approximately 2.71828).
• k! is the factorial of k.

Mean and Variance

Mean (Expected Value), E(X) = λ
Variance, Var(X) = λ

Observation: For a Poisson distribution, the mean and variance are equal.

Relationship with Binomial Distribution

The Poisson distribution can be derived as a limiting case of the Binomial distribution when:

• The number of trials (n) approaches infinity (n → &infty;).
• The probability of success (p) approaches zero (p → 0).
• The product n × p (which is the mean of the Binomial distribution) remains constant and equals λ (n × p → λ).

This means Poisson is suitable for events with a large number of opportunities but a small probability of occurrence.

Applications

• Number of calls received by a call center in an hour.
• Number of car accidents at a specific intersection in a week.
• Number of typos on a page of a book.
• Number of mutations in a given strand of DNA per unit length.

Example

A call center receives an average of 4 calls per minute. What is the probability that they receive exactly 2 calls in the next minute?

• λ = 4 (average calls per minute)
• k = 2 (number of calls we are interested in)

P(X = 2) = (4² × e^-4) / 2!

P(X = 2) = (16 × 0.018315) / 2

P(X = 2) = 0.14652

So, there is approximately a 14.65% chance of receiving exactly 2 calls in the next minute.

Normal Distribution

The Normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric about its mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is one of the most important distributions in statistics.

Definition

A continuous random variable X follows a Normal distribution if its probability density function forms a symmetric, bell-shaped curve. Many natural phenomena and statistical measures tend to follow this distribution.

Characteristics and Properties

• It is symmetric about its mean (μ). The curve's shape is identical on either side of the mean.
• The mean, median, and mode are all equal and located at the center of the distribution (μ = median = mode).
• The total area under the curve is equal to 1.
• The curve is asymptotic to the x-axis, meaning it approaches but never touches the x-axis.
• The shape of the normal distribution is determined by two parameters: its mean (μ) and its standard deviation (σ).
• Inflection points (where the curve changes from curving downward to curving upward) occur at μ - σ and μ + σ.

Parameters

• μ (mu): The mean of the distribution, which determines the center of the curve.
• σ (sigma): The standard deviation of the distribution, which determines the spread or width of the curve. A larger σ means a wider, flatter curve.

Probability Density Function (PDF)

The probability density function for a Normal distribution is given by:

f(x) = (1 / (σ × √(2π))) × e^{-(x-μ)2 / (2σ2)}

Where:

• x is any value of the continuous random variable.
• μ is the mean.
• σ is the standard deviation.
• π is pi (approximately 3.14159).
• e is Euler's number (approximately 2.71828).

Note: For continuous distributions, P(X = x) = 0. We always calculate probabilities over an interval, P(a ≤ X ≤ b), which corresponds to the area under the curve between a and b.

Standard Normal Distribution (Z-score)

A special case of the Normal distribution is the Standard Normal Distribution, which has a mean (μ) of 0 and a standard deviation (σ) of 1. Any Normal distribution can be transformed into a Standard Normal distribution using the Z-score formula:

Z = (X - μ) / σ

Where:

• Z is the Z-score, representing the number of standard deviations a data point is from the mean.
• X is the value from the original Normal distribution.
• μ is the mean of the original Normal distribution.
• σ is the standard deviation of the original Normal distribution.

Using Z-scores and standard normal tables allows us to find probabilities for any normal distribution.

Empirical Rule (68-95-99.7 Rule)

For a Normal distribution, approximately:

• 68% of the data falls within 1 standard deviation of the mean (μ ± 1σ).
• 95% of the data falls within 2 standard deviations of the mean (μ ± 2σ).
• 99.7% of the data falls within 3 standard deviations of the mean (μ ± 3σ).

Applications

• Modeling height, weight, blood pressure, or IQ scores in a population.
• Analyzing measurement errors in experiments.
• Financial modeling (e.g., stock price movements, returns).
• Quality control in manufacturing processes.

Example

The heights of adult men in a certain city are normally distributed with a mean (μ) of 175 cm and a standard deviation (σ) of 7 cm. What percentage of men have heights between 168 cm and 182 cm?

• μ = 175 cm
• σ = 7 cm

The range 168 cm to 182 cm is equivalent to (175 - 7) cm to (175 + 7) cm, which is μ ± 1σ.

According to the Empirical Rule, approximately 68% of the data falls within 1 standard deviation of the mean. Therefore, 68% of men have heights between 168 cm and 182 cm.

Exponential Distribution

The Exponential distribution is a continuous probability distribution that describes the time between events in a Poisson process, i.e., a process in which events occur continuously and independently at a constant average rate. It is often used to model the waiting time until the next event.

Definition

A continuous random variable X follows an Exponential distribution if it models the time elapsed between two successive events in a Poisson process. It is memoryless, meaning the probability of an event occurring in the future is independent of how much time has already passed.

Characteristics and Properties

• It is a continuous distribution for non-negative values (X ≥ 0).
• It is skewed to the right (positively skewed).
• The most notable property is the memoryless property: P(X > s+t | X > s) = P(X > t). This means that the probability of an event occurring in the future is independent of how much time has already passed. For example, if a machine has an exponentially distributed lifetime, the probability that it will last for another 't' hours is the same regardless of how long it has already been working.

Parameter

• λ (lambda): The rate parameter. It represents the average number of events per unit of time or space. It is the reciprocal of the mean time between events.

Probability Density Function (PDF)

The probability density function for an Exponential distribution is:

f(x) = λe^-λx for x ≥ 0
f(x) = 0 for x < 0

Where:

• x is the time elapsed.
• λ is the rate parameter.
• e is Euler's number.

Cumulative Distribution Function (CDF)

The cumulative distribution function, which gives the probability that an event occurs within time x, is:

F(x) = P(X ≤ x) = 1 - e^-λx for x ≥ 0
F(x) = 0 for x < 0

Mean and Variance

Mean (Expected Value), E(X) = 1 / λ
Variance, Var(X) = 1 / λ²

Observation: The mean time between events is the reciprocal of the rate parameter.

Relationship with Poisson Distribution

The Exponential distribution is closely related to the Poisson distribution. If the number of events in a given interval follows a Poisson distribution with rate λ, then the waiting time between these events follows an Exponential distribution with the same rate λ.

• Poisson: Counts the number of events in a fixed interval.
• Exponential: Measures the time until the next event (or between events).

Applications

• Modeling the lifetime of electronic components or devices.
• Analyzing the time until the next customer arrival at a service counter.
• Predicting the time between failures of a system.
• Measuring the waiting time for a bus or train.

Example

A particular type of light bulb has a lifetime that is exponentially distributed with a mean lifetime of 1000 hours. What is the probability that a bulb will last less than 500 hours?

• Mean lifetime E(X) = 1/λ = 1000 hours.
• Therefore, the rate parameter λ = 1/1000 = 0.001 events per hour.
• We want to find P(X ≤ 500).

Using the CDF: F(x) = 1 - e^-λx

P(X ≤ 500) = 1 - e^{-(0.001 × 500)}

P(X ≤ 500) = 1 - e^-0.5

P(X ≤ 500) = 1 - 0.60653

P(X ≤ 500) = 0.39347

So, there is approximately a 39.35% chance that a light bulb will last less than 500 hours.

Summary of Distributions