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 $P (X i = j)$ .

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 $X 1$ and $X 2$ 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 $X 1$ and $X 2$ .
A:
What should be the co-variance of $X 1$ and $X 2$ based on your understanding? Find the co-variance for the sample that you have.
A:
Estimate the joint distribution of $P (X 1, X 2)$ . From these estimates, do you think that the random variables are independent?
A:

Now generate 1000 samples of random variables $X 1$ and $X 2$ .

For $X 1$ , $p_1^0=p_1^1=0.5$ . For $X 2$ , $p (X 2 = 1 | X 1 = 1) = 0.7$ , $p (X 2 = 0 | X 1 = 1) = 0.3$ , $p (X 2 = 1 | X 1 = 0) = 0.3$ , $p (X 2 = 0 | X 1 = 0) = 0.7$ .

Find the co-variance of $X 1$ and $X 2$ .
A:
From the co-variance of $X 1$ and $X 2$ , do you think $X 1$ and $X 2$ are independent?
A:
Estimate the joint distribution of $P (X 1, X 2)$ . Do these estimates suggest that $X 1$ and $X 2$ are independent?
A:
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 $X 1$ , $X 2$ and $X 3$ . $X 1$ and $X 2$ are independent and identically distributed with $p_i^1=p_i^0=0.5$ for both $i = 1,2$ . $X 3$ is generated by flipping $X 2$ if $X 1$ is 1.

Is $X 3$ independent of $X 1$ ? Is $X 3$ independent of $X 2$ ? Find it out by estimating the joint probabilities.
A:
Are $X 1$ , $X 2$ and $X 3$ mutually independent i.e., is $P (X 1 = i, X 2 = j, X 3 = k) = P (X 1 = i) * P (X 2 = j) * P (X 3 = k)$ for all values of i,j and k? Try to answer it based on your understanding.
A:
Find the joint probability distribution of $P (X 1 = i, X 2 = j, X 3 = k)$ for all possible combination's of i,j and k. Does it suggest that $X 1, X 2 a n d X 3$ are independent?
A:

Lab 3 handouts

From SeedWiki

Lab 3: Independence and Identical Distribution

Single Random Variable Distribution

Two Random Variables: Distribution and Independence

Multiple Random Variable Independence

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