Probability Distribution
Distribution = An arrangement of values of a variable showing their frequency of occurrence
Each elementary event in a random experiment (sample point in the sample space) has a certain probability of occurrence. The random variable associates each event of the experiment with a numerical value. The set of possible numerical values related to an experiment forms the range of the random variable.
Therefore, each numerical value in the range has a certain probability of occurrence associated with it.
Eg: 
1. 
In the experiment of tossing 3 coins, where "H" represents a head on a coin and "T" a tail on a coin,
The sample space S = (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
If "s' represents the elements of the set "S" and "x" the number of heads obtained
Then, the random variable "X" representing the relationship between "s" and "x" is
X(s) = x (Or) X : s → x
The value of "x" as determined by the outcome of the experiment would be
Event [Coins Showing Up] 
HHH 
THH 
HTH 
HHT 
TTH 
THT 
TTH 
TTT 
"x" 
3 
2 
2 
2 
1 
1 
1 
0 
"X" is a function with domain "S" and Range = {0, 1, 2, 3}.
The number of elements in the sample space is 8 (i.e. n(s) =8) and all the events are all equally likely and mutually exclusive,
⇒ Probability of occurrence of each elementary event is 1/n i.e. 1/8
The probabilities of various events would be
Event [Coins Showing Up] 
HHH 
THH 
HTH 
HHT 
TTH 
THT 
TTH 
TTT 
P(Event) 








The probabilities of occurrence of the different values in the range of the random variable would be
P(X = 0) 
= 
P (TTT) 

⇒ P(X = 0) 
= 

P(X = 1) 
= 
P (HTT) + P(THT) + P(TTH) 
⇒ P(X = 1) 
= 

⇒ P(X = 1) 
= 

P(X = 2) 
= 
P (HHT) + P(HTH) + P(THH) 
⇒ P(X = 2) 
= 

⇒ P(X = 2) 
= 

P(X = 3) 
= 
P (HHH) 

⇒ P(X = 3) 
= 

P(X = 2) = P (HHT) + P(HTH) + P(THH)
Probability of getting 2 heads
⇒ Probability that at least one of these events should occur
⇒ P (HHT U HTH U THH) = P (HHT) + P(HTH) + P(THH)
["HHT", "HTH", "THH" are elementary events which are mutually exclusive,]
The distribution indicating the numerical values in the range of "X" and their respective probabilities of occurrence
This distribution indicating all the possible numerical values (within the range of a random variable in relation to an experiment) with their respective probabilities is called a probability distribution.

Discrete Probability Distribution
The distribution indicating the possible numerical values (within the range of a discrete random variable related to an experiment) and their respective probabilities is called a "Discrete Probability Distribution"
If "X" is a discrete random variable with range x_{1}, x_{2}, x_{3}, ...., x_{n} with respective probabilities p_{1}, p_{2}, p_{3}, ...., p_{n}, where p_{1} + p_{2} + p_{3} + ....+ p_{n} = 1, then the set P(X=x_{i}) = p(x_{i}) is called the probability distribution of the discrete random variable "X". It is generally represented as a table with the range of values of the discrete random variable and their respective probabilities together.
Eg: 
1. 
In the experiment of rolling a die,
The sample space S = (ONE, TWO, THREE, FOUR, FIVE, SIX}
The number of elements in the sample space is 6 (i.e. n(s) = 6) and all the events are all equally likely and mutually exclusive,
⇒ Probability of occurrence of each elementary event is 1/n i.e. 1/6
⇒ P(ONE) = P(TWO) = P(THREE) = P(FOUR) = P(FIVE) = P(SIX) = 1/6
If "s' represents the elements of the set "S" and "x" the number of heads obtained
Then, the random variable "X" representing the relationship between "s" and "x" is
X(s) = x (Or) X : s → x
The value of "x" as determined by the outcome of the experiment would be
This relationship can be summarised as
Event [Die Showing Up] 
ONE 
TWO 
THREE 
FOUR 
FIVE 
SIX 
"x" 
1 
2 
3 
4 
5 
6 
"X" is a function with domain "S" and Range = {1, 2, 3, 4, 5, 6}.
The probabilities of occurrence of the different values in the range of the random variable would be
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P(X = 1) 
= 
P (ONE) 
⇒ P(X = 1) 
= 

P(X = 2) 
= 
P (TWO) 
⇒ P(X = 2) 
= 

P(X = 3) 
= 
P (THREE) 
⇒ P(X = 3) 
= 

P(X = 4) 
= 
P (FOUR) 
⇒ P(X = 4) 
= 

P(X = 5) 
= 
P (FIVE) 
⇒ P(X = 5) 
= 

P(X = 6) 
= 
P (SIX) 
⇒ P(X = 6) 
= 

The distribution indicating the numerical values in the range of "X" and their respective probabilities of occurrence
This distribution indicating all the possible numerical values (within the range of a discrete random variable in relation to an experiment) with their respective probabilities is called a discrete probability distribution.


2. 
A person picking 3 balls from a bag can win Rs. 10 for each blue ball drawn and lose Rs. 5 for each red ball drawn.
In the experiment of drawing 3 balls from a bag containing 5 red and 4 blue balls, where "R" represents a red ball and "B" represents a blue ball
The sample space S = (BBB, BBR, BRB, RBB, RRB, RBR, BRR, RRR}
If "s' represents the elements of the set "S" and "x" the number of heads obtained
Then, the random variable "X" representing the relationship between "s" and "x" is
X(s) = x (Or) X : s → x
The value of "x" as determined by the outcome of the experiment would be
Event [Balls drawn being] 
BBB 
RBB 
BRB 
BBR 
RRB 
RBR 
RRB 
RRR 
"x" 
+ 30 
+ 15 
+ 15 
+ 15 
0 
0 
0 
− 15 
"X" is a function with Domain "S" and range = {−15, 0, +15 or +30}
The number of elements in the sample space is 8 (i.e. n(s) = 8) and all the events are all equally likely and mutually exclusive,
⇒ Probability of occurrence of each elementary event is 1/n i.e. 1/8
The probabilities of various events would be
Event [Coins Showing Up] 
HHH 
THH 
HTH 
HHT 
TTH 
THT 
TTH 
TTT 
P(Event)" 








The probabilities of occurrence of the different values in the range of the random variable would be
P(X = +30) 
= 
P (BBB) 

⇒ P(X = +30) 
= 

P(X = +15) 
= 
P(BBR) + P(BRB) + P(RBB) 
= 

⇒ P(X = +15) 
= 

P(X = 0) 
= 
P(BRR) + P(RBR) + P(RRB) 
= 

⇒ P(X = 0) 
= 

P(X = −15) 
= 
P (RRR) 

⇒ P(X = −15) 
= 

P(X = 0) = P (BRR) + P(RBR) + P(RRB)
Probability of getting a blue ball and 2 red balls
⇒ Probability that at least one of these events should occur
⇒ P (BRR U RBR U RRB) = P (BRR) + P(RBR) + P(RRB)
["BRR", "RBR", "RRB" are elementary events which are mutually exclusive, ]
"theory,expectation,theory,random,continous,discrete,variable,probability,distribution,mass,density,function,mean,variance,standard,deivation,"

The distribution indicating the numerical values in the range of "X" and their respective probabilities of occurrence
This distribution indicating all the possible numerical values (within the range of a discrete random variable in relation to an experiment) with their respective probabilities is called a discrete probability distribution.

