# 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

This lab is about distribution and independence. Download matlab files Lab3_1.m,Lab3_2.m,Lab3_3.m

Notation: I'll use $p_i^j$ 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
• A:
• Estimate the mean, variance and std-dev using the 1000 samples that we generated. Do the estimates match the calculated value?
• A:
• Use the samples to estimate the distribution i.e., find P(X = i) for i = − 1,0,1
• A:

### 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: $p_i^1=0.3$ and $p_i^0 = 0.7$ for both i = 1,2.

• From the sample, estimate $p_i^j$ for i = 1,2 and j = 0,1. Find the mean, variance, std-dev of X1 and X2.
• A:
• What should be the co-variance of X1 and X2 based on your understanding? Find the co-variance for the sample that you have.
• A:
• Estimate the joint distribution of P(X1,X2). From these estimates, do you think that the random variables are independent?
• A:

Now generate 1000 samples of random variables X1 and X2.

For X1, $p_1^0=p_1^1=0.5$. 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.
• A:
• From the co-variance of X1 and X2, do you think X1 and X2 are independent?
• A:
• Estimate the joint distribution of P(X1,X2). Do these estimates suggest that X1 and X2 are independent?
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• Is it always possible to estimate the joint distribution with sufficient accuracy? Think about the number of values the random variables can take.
• A:

### 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 $p_i^1=p_i^0=0.5$ 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.
• A:
• 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.
• A:
• 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?
• A: