Question

Given two sets of data having probability distributions (Normal distributions) with mean μ1, μ2 and variance σ1^2, σ2^2 respectively. How can we find the mean and variance of the data from both sets combined together as a single set? It is told that the distribution of the new combined set looks like multiplication of the two original probability distributions. Can anyone explain why and how?

Answer 1

lets test it out

x = {1,3,3,4,5,8,10}; mean x = 34/7

y = {3,12,15} mean y = 30/3

now are you asking about combining the sets? as in x union y?

x+y = {1,3,3,4,5,8,10,3,12,15}; mean x+y = (34+30)/(7+3) = 64/10

Answer 2

so, looks like multiplication, you are saying that the addition has some resemblence to multiplying fractions since its just a straight across process
\[\frac{34}{7}(+)\frac{30}{3}=\frac{34+30}{7+3}\]

Answer 3

combineing the sets ....

sum of x + sum of y = sum of combined sets

number of x + number of y = total number in both sets

the mean is sum divided by number of elements

Answer 4

Sorry that my explanation of the question was little vague.

Actual problem is as below,
Sensor A gives certain measurement data (say position of a robot arm or something) with certain accuracy or uncertainty (i.e. N(Mean1, Variance1)) and
Sensor B gives certain measurement data (i.e position of the same object mentioned above) with some accuracy (i.e. N(Mean2, Variance2))
Now i have to use both the data available from sensor A & B and find an appropriate value for the position of the object in discussion.

In simple meaning how to fuse two sensors' data to get a useful information?