GRE Data Analysis | Distribution of Data, Random Variables, and Probability Distributions

Distribution of Data:

• The median, represented by M, is between 730 and 740
• The mean, represented by m, is between 750 and 760
• The sum of areas of all 50 bars of relative frequency is 1

Random Variable:

S = { HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}

P(X = 1) is probability of occurring head one time,
P(X = 1) = P(THT) + P(TTH) + P(HTT) = 3/8

• Discrete Random Variable: A variable that can take one value from a discrete set of values. Example: Let x denote the sum of dice, Now x is discrete random variable as it can take one value from the set { 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }, since the sum of two dice can only be one of these values.
• Continuous Random Variable: A variable that can take one value from a continuous range of values. Example: x denotes the volume of water in a 500 ml cup. Now x may be a number from 0 to 500, any of which value, x may take.

Probability Distribution:

S = {HHH, HHT, HTH, THH, HTT, TTH, THT, TTT}

P(X = 0) = P(TTT) = 1/8
P(X = 1) = P(HTT) + P(TTH) + P(THT) = 3/8
P(X = 2) = P(HHT) + P(HTH) + P(THH) = 3/8
P(X = 3) = P(HHH) = 1/8 

1. Discrete distribution: Based on discrete random variable, examples are Binomial Distribution, Poisson Distribution.
2. Continuous distribution: Based on continuous random variable, examples are Normal Distribution, Uniform Distribution, Exponential Distribution.

Probability Mass Function: Let x be discrete random variable then its Probability Mass Function p(x) is defined such that

1. p(x) 0
2.  = 1
3. p(x) = P(X=x) 

Probability Density Function: Let x be continuous random variable then probability density function F(x) is defined such that

1. F(x)  0
2.  = 1
3. P(a < x < b) =  

Properties of Discrete Distribution:

1.  = 1
2. E(x) = 
3. V(x) =  

Properties of Continuous Distribution:

1.  = 1
2. E(x) = 
3. V(x) = 
4. p(a < x < b) =  

Where, E(x) denotes expected value or average value of the random variable x, V(x) denotes the variance of the random variable x. Types of Distributions:
• Uniform Distribution
• Exponential Distribution
• Normal Distribution
• Binomial Distribution
• Poisson Distribution