Discrete Probability distributions and joint probability distribution

Discrete Probability
Distribution
Probability function, Distribution function, Mean,
Variance, Problems

, MODULE I
DISCRETE PROBABILITY DISTRIBUTIONS

Defintions:

RANDOM EXPERIMENT: An experiment that can give different outcomes even though it is repeated
under same conditions is called random experiment

SAMPLE SPACE: The set of all possible outcomes of a random experiment is called sample space
denoted by S.

1. Tossing a coin has sample space S={ H, T}
2. Throwing a die S ={ 1, 2, 3, 4, 5, 6}
3. Tossing two coins simultaneously S = { HH, HT, TH, TT}
4. Tossing a coin twice S= { HH, HT, TH, TT}
5. Tossing a coin thrice S= { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
 Note: When a coin is tossed n times the no. of outcomes is 2n
 When a die is thrown n times the no. of outcomes is 6n


Prijimol V B, Assistant Professor, Christ college of Engineering ,
Irinjalakuda

,RANDOM VARIABLE: A variable that assigns a real number to every outcome of a random experimen
called a random variable

Random variables are denoted by capital letters X, Y, Z etc. and its values by small letters as x, y, z, …et

CLASSIFICATION OF RANDOM VARIABLES:

 Discrete random variable ( will learn in First module)
 Continuous random variable (will learn in Second module)

DISCRETE RANDOM VARIABLE: A random variable that takes only finite discrete number of value
called a Discrete random variable. It is usually an integer value

Eg: 1. No. of accidents on a road = 0, 1 , 2, 3….

2. No. of absentees in a class per day = 0, 1, 2, …..

CONTINUOUS RANDOM VARIABLE: A random variable that takes all possible values in an interva
called Continuous random variable. It is usually any real number in a interval.

Eg. Height, weight, temperature etc
Prijimol V B, Assistant Professor, Christ college of Engineering ,
Irinjalakuda

, PROBABILITY MASS FUNCTION (PMF) OR PROBABILITY FUNCTION OF A DISCRETE
RANDOM VARIABLE

The function P(X = x) = f(x) of a discrete random variable is said to be a PMF if it satisfies the following
two conditions
f(x) ≥ 0


 f ( x)  1


It is usually denoted by f(x) or p(x)

PROBABILITY DISTRIBUTION : The probability distribution is the set of values [ x , f(x) ]
CUMULATIVE DISTRIBUTION FUNCTION OR DISTRIBUTION FUNCTION (CDF) OF A
DISCRETE RANDOM VARIABLE
If X is a discrete random variable then its CDF is given by
F ( x)  P( X  x)   f (t )  Add all probabilities upto X  x
tx

It is usually denoted by F(x).

Mean=    xi pi
i 1

Variance=  2   xi 2 pi  2 Prijimol V B, Assistant Professor, Christ college of Engineering ,
i 1 Irinjalakuda