Sum of two Distributions

In sum of probability distributions and statistics, a probability will make the value of a random variable  or the probability of the value drawing  within a particular interval. The sum probability distribution defiines the measure of possible values like random variable can accomplish and the probability sum that the value of the random variable is within subset of that range.

Probability distributions:

Random Experiment and Sample Space

• Any experiment cannot be predicted in before, but is one of the set of possible outcomes, which is known as random experiment.
• An experiment performed frequently, every repetition is known as trial.
• The collection of all likely outcomes in random experiment is denoted the sample space.
• The sample space of a random experiment is denoted as a set S that includes all possible outcomes of the experiment
•  For simple experiments, the sample space may be exactly the collection  of possible outcomes.
• The sample space is a mathematically best set which contains the possible outcomes and perhaps other elements as well.
•  For example, if the testing is to draw a standard die and evidence the outcome, the sample space is S 1 2 3 4 5 6 , the set of possible outcomes.

Probability Distributions
      Let S acting as sample space with random experiment.
     Then X : x₁ x₂ x₃ …x_n
      P(X) is known as probability distribution of x.
Mean and variance of  Addition of two distributions:
Mean is E(X) = ∑ pi xi
variance is given below,
 

Examples:

1)Find the sum of mean of two distributions for the given below?

X	0	1	2
P(X)= x	0.25	.20	.25

X	0	1	2
P(X)= x	0.15	.20	.35

Solution:
Mean for table 1.
Mean E(X)1 = ∑ pi xi
            = 0*0.25 + 1*0.20+2*0.25
                       =0 + 0.20 +0.50  = 0.70
           Mean for table 2.
Mean E(X) 2 = ∑ pi xi
                    = 0*.15+1*0.20+2*0.35=0+.20+0.70
                 = 0.90
Sum of two distributions are E(X)1 + E(X)2= 0.70+0.90 = 1.6
2)Find out sum the variance of the following two distributions?

X	0	1	2
P(X)= x	0.25	.20	.25

X	0	1	2
P(X)= x	0.15	.20	.35