Q$2(60$ pts$)$: An e$-$commerce company is preparing to launch a new product and wants to

predict its success based on four factors: product category $($tech or not$),$ target demographic

$($young adults or not$),$ brand reputation $($high or not$),$ and advertising budget $($high or not$).$

The company has collected historical data from previous product launches and found the

following:

The probability of a product being successful $(S)$ depends on the brand reputation $(H_R)$

and the advertising budget $(H_B).-$ The product category $(T)$ and the target demographic

$(Y)$ are independent of each other. $-$ The brand reputation $(H_R)$ is influenced by both the

product category $(T)$ and the target demographic $(Y).-$ The brand reputation $(H_R)$ and

the advertising budget $(H_B)$ are independent of each other.

Here are the findings from the company's historical data:

$55\%$ of the products belong to the tech category.

$40\%$ of the products target young adults.

$65\%$ of the products have a high advertising budget.

$75\%$ of tech products targeting young adults have a high brand reputation; $50\%$ of

tech products targeting other demographics have a high brand reputation; $60\%$ of non$-$

tech products targeting young adults have a high brand reputation; $35\%$ of non$-$tech

products targeting other demographics have a high brand reputation.

$85\%$ of the products with high brand reputation and a high advertising budget are

successful; $55\%$ of the products with high brand reputation and a low advertising

budget are successful; $65\%$ of the products with low brand reputation and a high

advertising budget are successful; $25\%$ of the products with low brand reputation and

a low advertising budget are successful.

Questions:

Construct a Bayesian network that represents the relationships between the variables

based on the given scenario, and write the corresponding probability distribution for

each node. $($Hint: Remember to normalize the probability distribution so that the sum

is equal to $1.)[15$ points$]$

The company plans to launch a new non$-$tech product targeting a demographic other

than young adults with a high advertising budget. What is the probability that the

product will be successful? $[15$ points$]$

The company is considering two different product launch scenarios, both with a high

advertising budget: $($Scenario $1$:$)$ A tech product targeting young adults. $($Scenario

$2$:$)$ A non$-$tech product targeting a demographic other than young adults.

Compare the probabilities of success for these two scenarios. Which product has a

higher probability of success? $[20$ points$]$

Proof: