Explaining Probability Distribution: Types, solved examples.

In this post, we are understanding “What is Probability Distribution?”, “Types of Probability Distribution.” Along with definition, we will also solve some questions related to distribution.

It is important to know about Random Variable before to understand the Probability Distribution.

What is a Random Variable?

Random Variables play a vital role in probability distributions. In probability and statistics, a random variable is a variable whose possible values are the outcomes of a random phenomenon. It can be either discrete or continuous.

Let’s understand the Discrete and Continuous Random Variable.

Discrete Random Variable

If a variable can take the countable number of distinct values then it’s a discrete random variable.

For example: In an experiment of tossing 2 coins, we need to find out the possible number of heads. In this case, X is the random variable and the possible values taken by it is 0, 1 and 2 which is discrete.

Therefore X= {0, 1, 2}

Continuous Random Variable

A random variable is said to be continuous if it takes the infinite number of values in an interval.

For example: Suppose the temperature in a city lies between 30⁰ and 45⁰ centigrade. The temperature can take any value in the interval 30⁰ to 45⁰. So the temperature can be either 30.13⁰ or 40.15⁰, or it may be in 30.13⁰ or 40.15⁰.

What is Probability Distribution?

A probability Distribution is a statistical function which lists all the possible outcomes a random variable can take, in any random process, with its corresponding probability of occurrence.

Let X be the number of heads that result from the toss of 2 coins. Here X can take values 0,1, or 2. X is a discrete random variable.

The table below shows the probabilities associated with the different possible values of X. The probability of getting 0 heads is 0.25, 1 head is 0.50, 2 heads is 0.25.

Simple example of probability distribution given below based on above question.

Types of Different Probability Distributions

Discrete Probability Distribution

If the random variable can take only finite set of values (Discrete Random Variable), then its probability distribution is called as Discrete Probability Distribution (also known as Probability Mass Function or PMF). The Probability distribution of discrete random variables is the list of values of different outcomes and their respective probabilities. In other words, for a discrete random variable X, the value of the Probability Mass Function P(x) is given as,

P(x)= P(X=x)

If X, discrete random variable takes different values x1, x2, x3……

Binomial Distribution

Here we have Binomial Random Variable. A binomial random variable counts how often a particular event occurs in a fixed number of tries or trials. For a variable to be a binomial random variable, all the following conditions must be met:

• There are a fixed number of trials (a fixed sample size).
• On each trial, the event of interest either occurs or does not.
• The probability of occurrence (or not) is the same on each trial.
• Trials are independent of one another.

Examples of binomial random variables:

• Number of correct guesses at 30 true-false questions when you randomly guess all answers
• Number of winning lottery tickets when you buy 10 tickets of the same kind
• Number of left-handers in a randomly selected sample of 100 unrelated people

The probability distribution of a binomial random variable is called a binomial distribution.

Binomial Formula

Suppose a binomial experiment consists of n trials and results in x successes. If the probability of success on an individual trial is P, then the binomial probability is:

b(x; n, P) = _nC_x * P^x * (1 – P)^{n – x} 

Suppose we flip a coin two times and count the number of heads (successes). The binomial random variable is the number of heads, which can take on values of 0, 1, or 2. The binomial distribution is presented below.

Number of heads	Probability
0	0.25
1	0.50
2	0.25

The binomial distribution has the following properties:

• The mean of the distribution (μ_x) is equal to n * P .
• The variance (σ²_x) is n * P * ( 1 – P ).
• The standard deviation (σ_x) is sqrt[ n * P * ( 1 – P ) ].

For example, suppose you flip a fair coin 100 times and let X be the number of heads; then X has a binomial distribution with n = 100 and p = 0.50. Its mean is 

u=np=100(0.50)=50

Example

Suppose a dice is tossed 5 times. What is the probability of getting exactly 2 fours?

Solution: This is a binomial experiment in which the number of trials is equal to 5, the number of successes is equal to 2, and the probability of success on a single trial is 1/6 or about 0.167. Therefore, the binomial probability is:

b(2; 5, 0.167) = ₅C₂ * (0.167)² * (0.833)³ 
b(2; 5, 0.167) = 0.161

Poisson distribution Formula

Here, we have Poisson Discrete Variable.

Examples of the random variable that could be used in a Poisson probability distribution include the following:

• The number of cars that arrive at a tollbooth over a specific period of time (infinite/ continuous).
• The number of typographical errors found in a manuscript (infinite/ continuous)).

Poisson distribution is used under certain conditions. They are:

• The number of trials “n” tends to infinity
• Probability of success “p” tends to zero

The probability distribution of a Poisson random variable is called a Poisson distribution. The formula for the Poisson distribution function is given by:

f(x) =(e^{– λ} λ^x)/x!

Where,

e is the base of the logarithm, x is a Poisson random variable, λ is an average rate of value

In Poisson distribution, the mean is represented as E(X) = λ. 

For a Poisson distribution, the mean and the variance are equal. It means that E(X) = V(X)

Example: Some vehicles pass through a junction on a busy road at an average rate of 300 per hour. Find out the probability that none passes in a given minute?

Solution: First we will compute,

The average number of cars per minute is:

𝜇=300/60

𝜇 = 5

Applying the formula:

Mean and Variance of Poisson distribution.

If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ.

Continuous Probability Distribution

Unlike a discrete random variable, continuous random variable holds different values from an interval of real numbers. Probability distribution of continuous random variable is called as Continuous Probability Distribution (also known as Probability Density function or PDF).

Normal Distribution

The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also known as the Gaussian distribution and the bell curve.

The mean, median, and mode are all equal.

Let’s look at a pizza delivery example. Assume that a pizza restaurant has a mean delivery time of 30 minutes and a standard deviation of 5 minutes. Using the Empirical Rule, we can determine that 68% of the delivery times are between 25-35 minutes (30 +/- 5), 95% are between 20-40 minutes (30 +/- 2*5), and 99.7% are between 15-45 minutes (30 +/-3*5). The chart below illustrates this property graphically.

A continuous random variable Z is said to be a standard normal (standard Gaussian) random variable, shown as Z∼N(0,1), if its PDF is given by