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The document discusses probability distributions, explaining how they model the likelihood of outcomes for random variables. It differentiates between discrete and continuous probability distributions and outlines key properties, such as the requirement that probabilities must be non-negative and sum to one. Additionally, it covers concepts like random variables, expectation, variance, and specific types of distributions, including binomial and geometric distributions.
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0% found this document useful (0 votes)
17 views27 pagesProbability and Distribution
The document discusses probability distributions, explaining how they model the likelihood of outcomes for random variables. It differentiates between discrete and continuous probability distributions and outlines key properties, such as the requirement that probabilities must be non-negative and sum to one. Additionally, it covers concepts like random variables, expectation, variance, and specific types of distributions, including binomial and geometric distributions.
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
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Probability Distribution - Function, Formula, Table
Last Updated : 03 Dec, 2024
A probability distribution describes how the probabilities of different
outcomes are assigned to the possible values of a random variable. It
provides a way of modeling the likelihood of each outcome in a random
experiment.
While a frequency distribution shows how often outcomes occur in a
sample or dataset, a probability distribution assigns probabilities to
outcomes in an abstract, theoretical manner, regardless of any specific
dataset. These probabilities represent the likelihood of each outcome
occurring.
Probability Distribution oe
Uniform Exponential Normat
aa
u ae BSS
Binomial Geometric Hypergeometric
Lally. i | al ee
Number System and Arithmetic Algebra Set Theory Probability Statistics Geome
Ina discrete probability distribution, the random variable takes distinct
values (like the outcome of rolling a die). In a continuous probability
distribution, the random variable can take any value within a certain range
(like the height of a person).
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! wer 572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
Key properties of a probability distribution include:
* The probability of each outcome is greater than or equal to zero.
* The sum of the probabilities of all possible outcomes equals 1.
Also Read: Frequency Distribution
Table of Content
‘Types of Random Variables in Probability Distribution
Probability Distribution of a Random Variable
Expectation (Mean) and Variance of a Random Variable
Different Types of Probability Distributions
Cumulative Probability Distribution
Probability Distribution Function
Random Variables
Random Variable is a real-valued function whose domain is the sample
space of the random experiment. It is represented as X(sample space) =
Real number.
We need to learn the concept of Random Variables because sometimes we
are just only interested in the probability of the event but also the number
of events associated with the random experiment. The importance of
random variables can be better understood by the following example:
Why do we need Random Variables?
Let's take an example of the coin flips. We'll start with flipping a coin and
finding out the probability. We'll use H for ‘heads’ and T for ‘tails’
So now we flip our coin 3 times, and we want to answer some questions,
1. What is the probability of getting exactly 3 heads?
2. What is the probability of getting less than 3 heads?
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! 2er 572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
3. What is the probability of getting more than 1 head?
Then our general way of writing would be:
1. P(Probability of getting exactly 3 heads when we flip a coin 3
times)
2. P(Probability of getting less than 3 heads when we flip a coin 3
times)
3. P(Probability of getting more than 1 head when we flip a coin 3
times)
In a different scenario, suppose we are tossing two dice, and we are
interested in knowing the probability of getting two numbers such that
their sum is 6.
So, in both of these cases, we first need to know the number of times the
desired event is obtained i.e. Random Variable X in sample space which
would be then further used to compute the Probability P(X) of the event.
Hence, Random Variables come to our rescue. First, let's define what is
random variable mathematically.
Random Variable X
Real Number Line
Events in SS
Sample Space
A random variable is a real valued function whose domain is the
sample space of a random experiment
psa gekstorgeeks.orprobaiity-dstbon ant 112/25, 12:52 PM Probabilly Dstrbution- Function, Formula, Tale | GaaksforGeeks
To understand this concept in a lucid manner, let us consider the
experiment of tossing a coin two times in succession.
The sample space of the experiment is $ = {HH, HT, TH, TT}. Let's define
a random variable to count events of head or tails according to our need,
let X be a random variable that denotes the number of heads obtained. For
each outcome, its values are as given below:
X(HH) = 2, X (HT) = 1, X (TH) = 1, X (TT) = 0.
More than one random variable can be defined in the same sample space.
For example, let Y be a random variable denoting the number of heads
minus the number of tails for each outcome of the above sample space S.
Y(HH) = 2-0 = 2; Y (HT) =1-1=0;Y (TH) =1-1=0;Y(TT)=0
-2=-2
Thus, X and Y are two different random variables defined on the same
sample
Note: More than one event can map to same value of random
variable.
Types of Random Variables in Probability Distribution
There are following two types of Random Variables:
* Discrete Random Variables
* Continuous Random Variables
Discrete Random Variables in Probability Distribution
hitps www. geeksforgeeks.orp/probablty-distribution! arr 572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
A Discrete Random Variable can only take a finite number of values. To
further understand this, let's see some examples of discrete random
variables:
1. X = {sum of the outcomes when two dice are rolled}. Here, X can only
take values like {2, 3, 4, 5, 6...10, 11, 12}.
2. X = {Number of Heads in 100 coin tosses}. Here, X can take only integer
values from [0, 100].
Continuous Random Variable in Probability Distribution
A Continuous Random Variable can take infinite values in a continuous
domain. Let's see an example of a dart game.
Suppose, we have a dart game in which we throw a dart where the dart
can fall anywhere between [-1, 1] on the x-axis. So if we define our
random variable as the x-coordinate of the position of the dart, X can take
any value from [-1, 1]. There are infinitely many possible values that X can
take. (X = {0.1, 0.001, 0.01, 1, 2, 2.112121 .... and so on}.
Probability Distribution of a Random Variable
Now the question comes, how to describe the behavior of a random
variable?
Suppose that our Random Variable only takes finite values, like x1, x2, X3
and xp. i.e., the range of X is the set of n values is {x1, x2, X3 .... and Xp}.
Thus, the behavior of X is completely described by giving probabilities for
all the values of the random variable X
Event Probability
x1 PIK= x3)
x2, P(X = x2)
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! sar 112/25, 12:52 PM Probabilly Dstrbution- Function, Formula, Tale | GaaksforGeeks
Event Probability
x3 P(X = x3)
The Probability Function of a discrete random variable X is the
function p(x) satisfying.
P(x) = P(X = x)
Random Variable X
Events in
‘Sample Space
Let's look at an example:
Example: We draw two cards successively with replacement from a well-
shuffled deck of 52 cards. Find the probability distribution of finding aces.
Answer:
Let's define a random variable "X’, which means number of aces. So
since we are only drawing two cards from the deck, X can only take
three values: 0, 1 and 2. We also know that, we are drawing cards
with replacement which means that the two draws can be
considered an independent experiments.
P(X = 0) = P(both cards are non-aces
= P(non-ace) x P(non-ace)
= Me
=i
hitps www. geeksforgeeks.orp/probablty-distribution! ear 205, 1252 Pu Probab istruton Function, Fomul, Tele | Geskstereeks
P(X = 1) = Plone of the cards in ace)
= P(non-ace and then ace) + Place and then non-ace)
= P(non-ace) x P(ace) + Place) x P(non-ace)
a8 A psy
52% a + a2 ™ Ga io
P(X = 2) = P(Both the cards are aces)
Place) x Place)
say 4-4
aX 1
Now we have probabilities for each value of random variable. Since it
is discrete, we can make a table to represent this distribution. The
table is given below.
x
PIX=*) taan69 24/169 1/169
It should be noted here that each value of P(X = x) is greater than zero and
the sum of all P(X = x) is equal to 1.
Probability Distribution Formulas
The various formulas under Probability Distribution are tabulated below:
Types of Distribution Formula
P(X) = "Cya*b”™
Where a = probability of success
b=probability of failure
n= number of trials
Binomial Distribution
x=random variable denoting success
Fl) = [ofl (tat
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! mer 512125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGoeks
Types of Distribution Formula
P(x) = nlf rlin-r)!. pl(1-p)™
Discrete Probability Distribution renee
P(x) = C(nur) . p'(1-p)
Expectation (Mean) and Variance of a Random Variable
Suppose we have a probability experiment we are performing, and we
have defined some random variable(R.V.) according to our needs( like we
did in some previous examples). Now, each time an experiment is
performed, our R.V. takes on a different value. But we want to know that if
we keep on doing the experiment a thousand times or an infinite number
of times, what will be the average value of the random variable?
Expectation
The mean, expected value, or expectation of a random variable X is written
as E(X) orux. If we observe N random values of X, then the mean of the N
values will be approximately equal to E(X) for large N.
For a random variable X which takes on values x}, Xz, X3_, Xn with
probabilities p1, P2, p3.. Pn. Expectation of X is defined as,
4
Din bP
ie it is weighted average of all values which X can take, weighted by
the probability of each value.
To see it more intuitively, let’s take a look at this graph below,
Now in the above figure, we can see both the Random Variables have the
almost same ‘mean’, but does that mean that they are equal? No. To fully
describe the properties/behavior of a random variable, we need something
more, right?
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! arr ss, 1282 PM Probably Distrbuton — Functor, Foul, Tbe | GeckforGeeks
We need to look at the dispersion of the probability distribution, one of
them is concentrated, but the other is very spread out near a single value.
So we need a metric to measure the dispersion in the graph.
Variance
In Statistics, we have studied that the variance is a measure of the spread
or scatter in the data. Likewise, the variability or spread in the values of a
random variable may be measured by variance.
For a random variable X which takes on values xy, X2, X3... Xp with
probabilities p1, P2, P3.. Pn and the expectation is E[X]
The variance of X or Var(X) is denoted by, E|X-ul? = YX(2i — 2)*pz,
B(X*|-(B[X))?
Let's calculate the mean and variance of a random variable probability
distribution through an example:
Example: Find the variance and mean of the number obtained on a throw
of an unbiased die.
Answer:
We know that the sample space of this experiment is {1, 2, 3, 4, 5,
oF
Let's define our random variable X, which represents the number
obtained on a throw.
So, the probabilities of the values which our random variable can
take are,
P(1) = P(2) = P(3) = P(A) = P(5) = P(6) = 1/6
Therefore, the probability distribution of the random variable is,
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! eer 572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
x 1 2 3 4 5 6
Probabilities 1/6 1/6 6 1/6 1/6 1/6
E[X] = Sip sti
=1xi4 1 ta 46x2
SILK GHEX EHIME HAR EFEX ETOH G
o
2 Lad? bagtx dager
aR xbaextssrxt+erxt
Also, E[X?] = 12 x 14.2? x
a
o
Thus, Var(X) = E[X2] - (E[X))?
So, therefore mean is # and variance is #
Different Types of Probability Distributions
We have seen what Probability Distributions are, now we will see different
types of Probability Distributions. The Probability Distribution’s type is
determined by the type of random variable. There are two types of
Probability Distributions:
* Discrete Probability Distributions for discrete variables
* Cumulative Probability Distribution for continuous variables
We will study in detail two types of discrete probability distributions,
others are out of scope in class 12.
Discrete Probability Distributions
Discrete Probability Functions also called Binomial Distribution assume
a discrete number of values. For example, coin tosses and counts of events
are discrete functions. These are discrete distributions because there are
no in-between values. We can either have heads or tails in a coin toss.
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! sor ss, 1282 PM Probably Distrbuton — Functor, Foul, Tbe | GeckforGeeks
For discrete probability distribution functions, each possible value has a
non-zero probability. Moreover, the sum of all the values of probabilities
must be one. For example, the probability of rolling a specific number on a
die is 1/6. The total probability for all six values equals one. When we roll
a die, we only get either one of these values.
Bernoulli Trials and Binomial Distributions
When we perform a random experiment either we get the desired event or
we don't. If we get the desired event then we call it a success and if we
don't it is a failure. Let’s say in the coin-tossing experiment if the
occurrence of the head is considered a success, then the occurrence of the
tailis a failure
Each time we toss a coin or roll a die or perform any other experiment, we
call it a trial. Now we know that in our experiments coin-tossing trial, the
outcome of any trial is independent of the outcome of any other trial. In
each of such trials, the probability of success or failure remains constant.
Such independent trials that have only two outcomes usually referred to
as ‘success’ or ‘failure’ are called Bernoulli Trials.
Definition:
Trials of the random experiment are known as Bernoulli Trials, if they
are satisfying below given conditions :
Finite number of trials are required.
All trials must be independent.
Every trial has two outcomes : success or failure.
Probability of success remains same in every trial.
Let's take the example of an experiment in which we throw a die; throwing
a die 50 times can be considered as a case of 50 Bernoulli trials, where the
result of each trial is either success(let’s assume that getting an even
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! ser ss, 1252 PM Probability Osrbuion Function, Famul, Table | GesksforGooks
number is a success) or failure( similarly, getting an odd number is failure)
and the probability of success (p) is the same for all 50 throws. Obviously,
The successive throws of the die are independent trials. If the die is fair
and has six numbers 1 to 6 written on six faces, then p = 1/2 is the
probability of success, and q = 1 - p =1/2 is the probability of failure.
Example: An urn contains 8 red balls and 10 black balls. We draw six
balls from the urn successively. You have to tell whether or not the trials of
drawing balls are Bernoulli trials when after each draw, the ball drawn is:
1. Replaced
2. Not replaced in the urn.
Answer:
1. We know that the number of trials are finite. When drawing is
done with replacement, probability of success (say, red ball) is p
=8/18 which will be same for all of the six trials. So, drawing of
balls with replacements are Bernoulli trials.
N
. If drawing is done without replacement, probability of success
(ie., red ball) in the first trial is 8/18, in 2nd trial is 7/17 if first
ball drawn is red or, 10/18 if first ball drawn is black, and so on.
Clearly, probabilities of success are not same for all the trials,
Therefore, the trials are not Bernoulli trials.
Binomial Distribution
It is a random variable that represents the number of successes in “N"
successive independent trials of Bernoulli's experiment. It is used in a
plethora of instances including the number of heads in “N” coin flips, and
soon
Let P and Q denote the success and failure of the Bernoulli Trial
respectively. Let's assume we are interested in finding different ways in
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! san? 572125, 1252 PM Probably Distrbution - Function, Formula, Table | GeoksforGeeks
which we have 1 success in all six trials.
Six cases are available as listed below:
PQQQAA, QPQQA, QQPAQ, QQQPAQ QQaagrg, aQagar
Likewise, 2 successes and 4 failures will show ;,combinations thus
making it difficult to list so many combinations. Henceforth, calculating
probabilities of 0, 1, 2,.., n number of successes can be long and time-
consuming. To avoid such lengthy calculations along with a listing of all
possible cases, for probabilities of the number of successes in n-
Bernoulli's trials, a formula is made which is given as:
IfY is a Binomial Random Variable, we denote this Y~ Bin(n, p), where p is
the probability of success in a given trial, q is the probability of failure, Let
‘n’ be the total number of trials, and ‘x’ be the number of successes, the
Probability Function P(Y) for Binomial Distribution is given as:
PLY) ="C, fp
where x = 0,1,2...n
Example: When a fair coin is tossed 10 times, find the probability of
getting:
1. Exactly Six Heads
2. At least Six Heads
Answer:
Every coin tossed can be considered as the Bernoulli trial. Suppose X
is the number of heads in this experiment:
We already know, n = 10
p=1/2
So, P(X = x) = "Cy pl (1-p)¥, X= 0,1,2,3,.nN
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! s927 wes oF rama once —Prter Foes Tenure
P(X = x) = °C p?*(1-pp*
When x = 6,
(i) Pix = 6) = 1°Cg p* (1-p)®
ax 4
xdx
(ii) Pat least 6 heads) = P(X >= 6) = P(X = 6) + P(X=7) + P(X=8)+
P(X=9) + P(X=10)
= 166 pt (1 — pl? + °C 7p? (1 pl” + 1°Cgp? (1 —pP? + °Cop"(1 — pI?
+7°C10 (1 —p)??
=101
aa
(aot
ae +8 7
Aw (2) 4 w(Ay 4
Tor?) ao!
out + iol
Negative Binomial Distribution
In a random experiment of discrete range, it is not necessary that we get
success in every trial. If we perform the ‘n’ number of trials and get
success ‘r’ times where n > r, then our failure will be (n ~ r) times. The
probability distribution of failure in this case will be called negative
binomial di:
ribution.
For example, if we consider getting 6 in the die is success and we want 6
one time, but 6 is not obtained in the first trial then we keep throwing the
die until we get 6. Suppose we get 6 in the sixth trial then the first 5 trials
will be failures and if we plot the probability distribution of these failures
then the plot so obtained will be called a negative binomial distribution.
Poisson Probability Distribution
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! sar ss, 1252 PM Probability Osrbuion Function, Famul, Table | GesksforGooks
The Probability Distribution of the frequency of occurrence of an event
over a specific period is called Poisson Distribution. It tells how many
times the event occurred over a specific period. It counts the number of
successes and takes a value of the whole number ie. (0,1,2..). It is
expressed as
f(x; A) = P(X = x) = (MeAl/x!
where,
* xis number of times event occurred
* e=2718..
* Ais mean value
Binomial Distribution Examples
Binomial Distribution is used for the outcomes that are discrete in nature.
Some of the examples where Binomial Distribution can be used are
mentioned below:
* To find the number of good and defective items produced by a factory.
* To find the number of girls and boys studying in a school.
* To find out the negative or positive feedback on something
Cumulative Probability Distribution
The Cumulative Probability Distribution for continuous variables is a
function that gives the probability that a random variable takes on a value
less than or equal to a specified point. It's denoted as F(x), where x
represents a specific value of the random variable. For continuous
variables, F(x) is found by integrating the probability density function (pdf)
from negative infinity to x. The function ranges from 0 to 1, is non-
decreasing, and right-continuous. It’s essential for computing probabilities,
ntips:iwwn.geokstorgeeks orgprobabily-aistrbution! sse7 ss, 1262 PM Probably Distibution Function, Formula, Table | GesksfrGeoks
determining percentiles, and understanding the behavior of continuous
random variables in various fields.
Cumulative Probability Distribution takes value in a continuous range; for
example, the range may consist of a set of real numbers. In this case,
Cumulative Probability Distribution will take any value from the continuum
of real numbers unlike the discrete or some finite value taken in the case of
Discrete Probability distribution. Cumulative Probability Distribution is of
two types, Continuous Uniform Distribution, and Normal Distribution.
Continuous Uniform Distribution
Continuous Uniform Distribution is described by a density function that is
flat and assumes value in a closed interval let's say [P, Q] such that the
probability is uniform in this closed interval. It is represented as f(x; P, Q)
f(x; P, Q) = 1Q- P) forPSx5Q
f(x; P, Q) = 0; elsewhere
Normal Distribution
Normal Distribution of continuous random variables results in a bell-
shaped curve. Itis often referred to as Gaussian Distribution by the name
of Karl Friedrich Gauss who derived its equation. This curve is frequently
used by the meteorological department for rainfall studies. The Normal
Distribution of random variable X is given by
nx; B, @) = {1/(V2ro}el-V/20*20CH"2 for -0