Lab 3 handouts
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Lab 3: Independence and Identical Distribution
This handout is available at http://seed.ucsd.edu/mediawiki/index.php/Lab_3_handouts. You can also go there from course webpage: Course Web Page -> Handouts -> Lab 3 Handout
Notation: I'll use to represent P(Xi = j).
Single Random Variable Distribution
We generate 1000 samples of random variable X. X takes on values -1,0,1 with probability 0.2, 0.7 and 0.1. Code is in matlab file Lab3_1.m
- Calculate the mean, variance and std-dev of X
- Estimate the mean, variance and std-dev using the 1000 samples that we generated. Do the estimates match the calculated value?
- Use the samples to estimate the distribution i.e., find P(X = i) for i = − 1,0,1
Two Random Variables: Distribution and Independence
Code for this part is in matlab file Lab3_2.m
We generate 1000 samples of random variables X1 and X2 which can take values 0 or 1. They are independent and identically distributed: and for both i = 1,2.
- From the sample, estimate for i = 1,2 and j = 0,1. Find the mean, variance, std-dev of X1 and X2.
- What should be the co-variance of X1 and X2 based on your understanding? Find the co-variance for the sample that you have.
- Estimate the joint distribution of P(X1,X2). From these estimates, do you think that the random variables are independent?
Now generate 1000 samples of random variables X1 and X2.
For X1, . For X2, p(X2 = 1 | X1 = 1) = 0.7, p(X2 = 0 | X1 = 1) = 0.3, p(X2 = 1 | X1 = 0) = 0.3, p(X2 = 0 | X1 = 0) = 0.7.
- Find the co-variance of X1 and X2.
- From the co-variance of X1 and X2, do you think X1 and X2 are independent?
- Estimate the joint distribution of P(X1,X2). Do these estimates suggest that X1 and X2 are independent?
- Is it always possible to estimate the joint distribution with sufficient accuracy? Think about the number of values the random variables can take.
Multiple Random Variable Independence
In the above example we were just judging independence between only 2 random variables. We now move to independence of more than 2 variables.
Code for this part is matlab file Lab3_3.m
Now we generate 10000 samples of 3 random variables X1, X2 and X3. X1 and X2 are independent and identically distributed with for both i = 1,2. X3 is generated by flipping X2 if X1 is 1.
- Is X3 independent of X1? Is X3 independent of X2? Find it out by estimating the joint probabilities.
- Are X1, X2 and X3 mutually independent i.e., is P(X1 = i,X2 = j,X3 = k) = P(X1 = i) * P(X2 = j) * P(X3 = k) for all values of i,j and k? Try to answer it based on your understanding.
- Find the joint probability distribution of P(X1 = i,X2 = j,X3 = k) for all possible combination's of i,j and k. Does it suggest that X1,X2andX3 are independent?