Probability Distribution

With the help of these experiments or events, we can always create a probability pattern table in terms of variables and probabilities. For additional educational resources

Probability distribution of random variables

A random variable has a probability distribution that specifies the probability of its unknown values. Random variables can be discrete (not constant) or continuous, or both. That means it takes any of a designated finite or countable list of values, provided with a probability mass function feature of the random variable’s probability distribution, or can take any numerical value in an interval or set of intervals.

Two random variables with equal probability distribution can yet be different in their relationships with other random variables or whether they are independent of these.

Detecting a random variable means that the outcomes of randomly selecting values based on the variable probability distribution function are called random variables.

Types of the probability distribution

There are two types of probability distributions that are used for different purposes and different types of data production processes.

1. Normal or cumulative probability distribution
2. Binomial or discrete probability distribution

Cumulative probability distribution:

Cumulative probability distributions are also known as continuous probability distributions. In this distribution, a set of possible outcomes can take values in a continuous range. For example, a set of real numbers, is a continuous or normal distribution, as it gives all the possible outcomes of real numbers.

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The formula for the normal distribution is:

\(\color{blue}{P\left(x\right)=\frac{1}{\sigma \sqrt{2\pi }}e^{-\frac{1}{2}\left(\frac{x-\mu }{\sigma }\right)^2}}\)

Where,

• \(\mu =\) Mean value
• \(σ =\) Standard distribution of probability
• \(x =\) Normal random variable

If mean \((μ) = 0\) and standard deviation \((σ) = 1\), then this distribution is known to be a normal distribution.

Discrete probability distribution:

A distribution is called a discrete probability distribution, in which a set of outcomes is discrete in nature. For example, if a dice is rolled, then all the possible outcomes are discrete and give a mass of outcomes. It is also known as the probability mass function.

Therefore, the outcomes of a binomial distribution include n repeated experiments, and the outcome may or may not occur. The formula for the binomial distribution is:

\(\color{blue}{P\left(x\right)=\frac{n!}{r!\left(n-r\right)!}.p^r\left(1-p\right)^{n-r}}\)

\(\color{blue}{P\left(x\right)=C\:\left(n,\:r\right).p^r\left(1-p\right)^{n-r}}\)

Where,

• \(n =\) Total number of events
• \(r =\) Total number of successful events
• \(p =\) Success on a single trial probability
• \(^nC_r=\left[\frac{n!}{r!\left(n-r\right)!}\right]\)
• \(1 – p =\) Failure Probability

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Frequently Asked Questions

What is a probability distribution?

A probability distribution is a function that assigns probabilities to the possible values of a random variable. For a discrete variable, it’s a table or formula giving the probability of each outcome. For a continuous variable, it’s a curve whose area under any interval equals the probability the variable lands in that interval.

What’s the difference between discrete and continuous distributions?

Discrete random variables can take only specific values — the number of heads in 10 coin flips, for example. Continuous random variables can take any value in an interval — height, time, temperature. Discrete distributions use a probability mass function (PMF); continuous distributions use a probability density function (PDF).

What is a PMF?

A probability mass function for a discrete variable \( X \) gives \( P(X = x) \) for each possible value \( x \). Two rules: \( 0 \le P(X=x) \le 1 \) for every \( x \), and the sum of \( P(X=x) \) over all \( x \) equals 1.

What is a PDF?

A probability density function for a continuous variable \( X \) is a curve \( f(x) \) such that the probability \( X \) falls in an interval \( [a, b] \) equals the area under \( f \) from \( a \) to \( b \). Note: for a continuous variable, \( P(X = x) \) is always 0 — only intervals have non-zero probability.

How do I find the expected value of a discrete distribution?

Multiply each value by its probability, then add the products: \( E[X] = \sum x \cdot P(X = x) \). For a single roll of a fair six-sided die: \( E[X] = (1+2+3+4+5+6)/6 = 3.5 \).

What is variance and how is it computed?

Variance measures how far values typically fall from the mean. For a discrete distribution: \( \text{Var}(X) = \sum (x – \mu)^2 \cdot P(X = x) \), where \( \mu = E[X] \). The square root of variance is the standard deviation, which is in the same units as \( X \).

What is the binomial distribution?

The binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. If you flip a coin 10 times, the number of heads follows a binomial distribution with \( n = 10 \) and \( p = 0.5 \). Its formula is \( P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \).

What is the normal distribution?

The normal (or Gaussian) distribution is a continuous, bell-shaped distribution defined by its mean and standard deviation. It models many real-world quantities — heights, measurement errors, test scores. About 68% of values lie within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3 — the empirical (\”68-95-99.7\”) rule.

What is the uniform distribution?

A uniform distribution gives equal probability to every value in its range. A fair die is discrete uniform — \( P(X=k) = 1/6 \) for \( k = 1, \ldots, 6 \). The continuous uniform distribution on \( [a, b] \) has constant density \( 1/(b-a) \); the probability of any sub-interval is just its length divided by \( b – a \).

Where do probability distributions show up in real life?

Almost everywhere there’s uncertainty: quality control (binomial for defect counts), modeling rare events (Poisson for accidents per day), measurement error (normal), insurance pricing (long-tailed distributions), genetics (binomial in inheritance), and finance (log-normal for stock returns). Once you can read the distribution, the rest of the analysis is mostly arithmetic.

Related Lessons You May Like

• How to find the probability of an event
• How to solve probability problems
• How to find the measures of central tendency
• How to find experimental probability
• How to solve conditional and binomial probabilities

If you want a friendly introduction to probability and statistics that builds the intuition before the formulas, Pre-Algebra for Beginners covers the prerequisites and includes a probability chapter. For the deeper distributions, expected value, and combinatorics, Algebra II for Beginners goes further.