Probability Distribution Function

Probability Distribution refers to the function that gives the probability of all possible values of a random variable.It shows how the probabilities are assigned to the different possible values of the random variable.
Common types of probability distributions Include:

• Binomial Distribution.
• Bernoulli Distribution.
• Normal Distribution.
• Geometric Distribution.

Note: A Probability Distribution can also be seen as the set of all possible outcomes of a random experiment, showing how probabilities are distributed across the values of the random variable.

A Probability Distribution Function (PDF) is a mathematical function that describes the likelihood of different outcomes in a random experiment. For any random variable X, where its value is evaluated at the points 'x', then the probability distribution function gives the probability that X takes the value less than equal to x.
We represent the probability distribution as, F(x) = P (X ≤ x)

Probability Distribution Function is also called Cumulative Distribution Function(CDF), The CDF represents the cumulative probability up to a certain value of the random variable.
The cumulative probability for a closed interval(a, b] is given by:

P(a < X ≤ b) = F(b) – F(a)

Note: For probability distribution function the value of the variable lies between 0 and 1: 0 ≤ F(x) ≤ 1

Related Reads: Probability Distribution

Probability Distribution Function (PDF) Formula

The probability distribution function formula gives the probability of all the possible outcomes of any random variable. Depending on the type of random variable (discrete or continuous), the formulas for the probability distribution function can differ.

Different formulas for the probability distribution function are listed beolw:

Probability Distribution of a Discrete Random Variable

Discrete Random Variable is the variable that takes distinct countable values are 0, 1, 2, 3 ... The formula for the probability distribution of a discrete random variable is,

Probability Distribution Function:

F(x) = P (X ≤ x)

Probability Distribution of a Continuous Random Variable

A continuous Random Variable is a variable that takes infinitely many values. The formula for the probability distribution of a continuous random variable is,
Probability Distribution Function: F(x) = P (X ≤ x)

Additionally, the Probability Density Function (PDF) is the derivative of the CDF:

f(x) = d/dx (F(x))

where,

• F(x) = ∫^x_-∞ f(u)du

Normal Probability Distribution Formula

The Normal Distribution, also known as the Gaussian distribution, is commonly represented by a bell-shaped curve. The formula for a normal probability distribution is:

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

where,

• μ is the Mean
• σ is the Standard Distribution
• x is the Normal random variable

Note: If mean(μ) = 0 and standard deviation(σ) = 1, then this distribution is called the Normal Distribution.

Binomial Probability Distribution Formula

The Binomial Distribution describes the probability of exactly r successes in n independent trials, where each trial has a probability p of success and 1 − p of failure. The formula for the binomial probability distribution is:

P(r out of n) = n!/r!(n − r)! × p^r(1 − p)^{n - r} = ⁿC_r × p^r(1 − p)^n−r

where,

• n is the Total number of events
• r is the Total number of successful events
• p is the Probability of success on a single trial
• 1 – p is the Probability of failure

Note: The binomial coefficient(ⁿ_r) is given by: ⁿC_r =  n!/r!(n – r)!

Probability Distribution Graph

The graph that plot the Probability Distribution Functions are called the Probability Distribution graphs. These graphs help us to visualize the probability distribution around a random variable and help us to easily find the required solution.

The sum of all the probabilities in any discrete distribution is one and for a continuous distribution of random variables the area under the graph is equal to 1. The distribution graph of the continuous distribution function is added below, where X (the random variable) lies between a and b. It is made using the Probability Density Function

For discrete random variables, the probability distribution is given using the Bernoulli distribution.

Probability Distribution Function and Probability Density Function

We easily describe the Probability distribution using the Probability Distribution Function and Probability Density Function. Using a probability distribution function is very useful for both continuous probability distribution and discrete probability distribution, while the probability density function(pdf) is only used for continuous probability distribution.

Articles related to Probability Distribution Function:

• Poisson Distribution
• Bernoulli Trials
• Binomial Distribution

Uses of Probability Distribution Function

• Statistical Inference: PDFs are fundamental in statistical inference, allowing for the estimation of population parameters and hypothesis testing.
• Modeling and Simulations: PDFs are used to model real-world phenomena and to simulate random processes in fields like engineering, finance, and the natural sciences.
• Risk Assessment: In finance and insurance, PDFs help assess risks and determine the likelihood of various financial outcomes.

Probability Distribution Function Examples

Example 1: Suppose we toss two dice. Make a table of the probabilities for the sum of the dice. The possibilities are: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12.

Solution:

Probability Distribution Table

X	P(x)
2	1/36
3	2/36
4	3/36
5	4/36
6	5/36
7	6/36
8	5/36
9	4/36
10	3/36
11	2/36
12	1/36

Example 2: The number of old people living in houses on a randomly selected city block is described by the following probability distribution.

What is the probability that 6 or more old people live in a randomly selected house?

Solution:

Sum of all the p(probability) is equal to 1

Probability that six or more old peoples live in a house,

= 1 - (0.50 + 0.25 + 0.10)
= 0.15

Thus, probability that six or more old peoples live in a house is equal to 0.15

Example 3: When a fair coin is tossed 8 times, then the Probability of:

• Exactly Four Heads
• At least Four Heads

Solution:

Every coin tossed can be considered as the Bernoulli trial. Suppose X be the number of heads in this experiment,

n = 8
p = 1/2

So,
P(X = x) = ⁿC_x p^{n - x} (1 - p)^x, x =  0, 1, 2, 3,...n
P(X = x) = ⁸C_xp^{8 - x}(1 - p)^x

P(Exactly 4 Heads)
= P(x = 4)
= ⁸C₄ p⁴ (1 - p)⁴
= 8!/4!4!(1/2)⁴(1/2)⁴
= (8 × 7 × 6 × 5/2 × 3 × 4) × (1/16) × (1/16)
= 420/1536
= 35/128

Thus, the probability of Exactly Four Heads in a Eight Coin Toss experiment is 35/128

P(At Least 4 Heads)

= P(X >= 4)
= P(X = 4) + P(X = 5) + P(X = 6)+ P(X = 7) + P(X = 8).
= ⁸C₄ p⁴ (1 - p)⁴ + ⁸C₅ p³ (1 - p)⁵ + ⁸C₆ p² (1 - p)⁶ + ⁸C₇ p¹(1 - p)⁷ + ⁸C₈(1 - p)⁸
= 8!/4!4!(1/2)⁸ + 8!/5!3!(1/2)⁸ + 8!/6!2!(1/2)⁸ + 8!/7!1!(1/2)⁸ + 8!/8!(1/2)⁸
= 8 × 7 × 6 × 5/4 × 3 × 2 × 256 + 8 × 7 × 6/3 × 2 × 256 + 8/256 + 1/256
= 1680/6144 + 336/1536 + 9/256
= 70/256 +  56/256 + 9/256
= 135/256

Thus, the probability of Atleast Four Heads in a Eight Coin Toss experiment is 135/256

Example 4: Calculate the probability of getting 10 heads, if a coin is tossed 12 times.

Solution:

Given,

• Number of Trials(n) = 12
• Number of Success(r) = 10 (getting 10 heads)
• Probability of Single Head(p) = 1/2 = 0.5

To find ⁿC_r =  n!/r!(n – r)!

=  12!/10!(12 – 10)!
=  (12 × 11 × 10!)/10!2!
= 66

To find p^r = (0.5)¹⁰ = 0.00097665625
So, the probability of getting 10 heads is:

P(x) = ⁿC_r p^r (1 - p)^{n - r}
= 66 × 0.00097665625 × (1 – 0.5)^(12-10)
= 0.0644593125 × (0.5)²
= 0.016114828125

The probability of getting 10 heads = 0.0161...

Example 5: Suppose that each time you take a free throw shot, you have a 35% chance of making it. If you take 25 shots, what is the probability of making exactly 15 of them?

Solution:

Given,

• n = 25
• r = 15
• p = 0.35
• q = 0.65

Compute
C_25,15 (0.35)¹⁵ (0.65)¹⁰ = 0.165

There is a 16.5% chance of making exactly 15 shots.

Example 6: There is a total of 5 people in the room, what is the possibility that someone in the room shares His / Her birthday with at least someone else?

Solution:

P(s) = p(At least someone shares with someone else)                        

P(d) = p(No one share their birthday everyone has a different birthday)

p(s) + p(d) = 1 or 100%
p(s) =100% - p(d)

There are 5 people in the room, the possibility that no one shares his/her birthday

= (365 × 364 × 363 × 362 × 361) ⁄ (365)⁵
= (365! ⁄ (365 - 5)!) ⁄ 365⁵
= (365! ⁄ 360!) ⁄ 365⁵
= 0.9728

p(d) = 0,9728 or 97.28%

p(s) = 100% - p(d)

= 100% - 97.28% or 1 - 0.9728
= 2.72%  ≈ 0.0272

Practice Questions on the Probability Distribution Function

Q1: Find the Probability Distribution of the the Number of Heads when two coins are tossed Simultaneously.

Q2: What is the Probability Distribution of the of number of Kings when three cards are drawn at random.

Q3: A die is thrown twice. Find the probability of getting a number of sixes.

Q4: A coin is thrown until a tail appears or the the head appears three times continuously. Find the probability distribution of tosses.