Bayesian Networks have shaped complicated issues with few knowledge and

resources. It is incorporated into the most cutting$-$edge technologies of the

modern day, including Articial Intelligence and Machine Learning.

You have three coins in your pocket:

$1.$ Coin $1$ is a fair coin that comes up heads with probability $1/2.$

$2.$ Coin $2$ is a biased coin that comes up heads with probability $1/4.$

$3.$ Coin $3$ is a biased coin that comes up heads with probability $3/4.$

Question $1$:

Suppose X and Y are independent random variables over the domain $1$; $2$; $3$

with P$($X $=3)=1/6.$ Given the following partially specied joint distribution, what are the remaining values? Write your answers as simplied

fractions in the blanks.

P$($X $=1$; Y $=1)=1/4$ P$($X $=2$; Y $=1)=1/6$ P$($X $=3$; Y $=1)=??$

P$($X $=1$; Y $=2)=1/16$ P$($X $=2$; Y $=2)=1/24$ P$($X $=3$; Y $=2)=??$

P$($X $=1$; Y $=3)=??$ P$($X $=2$; Y $=3)=??$ P$($X $=3$; Y $=3)=??$