12th Com Maths Part 2 Chapter 8 (Digest) Maharashtra state board

Chapter 8 Probability distributions

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Probability distributions are fundamental concepts in mathematics and statistics, describing how probabilities are distributed over the possible values of a random variable. They provide a mathematical function that gives the probabilities of occurrence of different possible outcomes. Probability distributions can be classified into two main types: discrete and continuous.

1. Discrete Probability Distributions

Discrete probability distributions deal with variables that take on a finite or countable number of values. Examples include the number of heads in a series of coin flips or the number of students present in a classroom. Some common discrete distributions include:

• Binomial Distribution: Describes the number of successes in a fixed number of independent Bernoulli trials (each trial has two possible outcomes, typically called "success" and "failure"). The probability mass function (PMF) is:

  $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

  where $n$ is the number of trials, $k$ is the number of successes, and $p$ is the probability of success on each trial.

• Poisson Distribution: Describes the number of events occurring within a fixed interval of time or space, given that these events happen at a constant rate and independently of the time since the last event. The PMF is:

  $$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$$

  where $\lambda$ is the average number of events in the interval.

2. Continuous Probability Distributions

Continuous probability distributions deal with variables that take on an infinite number of values within a given range. Examples include the height of people or the time it takes to run a race. Some common continuous distributions include:

• Normal Distribution: Also known as the Gaussian distribution, it is characterized by its bell-shaped curve and is defined by its mean (µ) and standard deviation (σ). The probability density function (PDF) is:

  $$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

• Exponential Distribution: Describes the time between events in a Poisson process. It is often used to model waiting times. The PDF is:

  $$f(x) = \lambda e^{-\lambda x}$$

  where $\lambda$ is the rate parameter.

• Uniform Distribution: All outcomes are equally likely within a given interval [a, b]. The PDF is:

  $$f(x) = \frac{1}{b - a} \quad \text{for} \quad a \leq x \leq b$$

Key Concepts in Probability Distributions

• Probability Mass Function (PMF): Used for discrete variables, it provides the probability that a discrete random variable is exactly equal to some value.

• Probability Density Function (PDF): Used for continuous variables, it describes the relative likelihood for this random variable to take on a given value.

• Cumulative Distribution Function (CDF): Describes the probability that a random variable will be less than or equal to a certain value. For a random variable $X$, the CDF is defined as:

  $$F(x) = P(X \leq x)$$

• Expectation (Mean): The long-run average value of repetitions of the experiment it represents. For a discrete random variable $X$:

  $$E(X) = \sum_{i} x_i P(X = x_i)$$

  For a continuous random variable $X$:

  $$E(X) = \int_{-\infty}^{\infty} x f(x) \, dx$$

• Variance: Measures the spread of the random variable around the mean. For a discrete random variable $X$:

  $$\text{Var}(X) = E[(X - E(X))^2] = \sum_{i} (x_i - E(X))^2 P(X = x_i)$$

  For a continuous random variable $X$:

  $$\text{Var}(X) = \int_{-\infty}^{\infty} (x - E(X))^2 f(x) \, dx$$

Probability distributions are essential tools in various fields such as statistics, finance, science, and engineering, providing a framework for modeling uncertainty and making informed decisions based on probabilistic data.