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Evaluate the log likelihood function for the data X=45, label (Y)=0, predicted probability is 0.95.

To evaluate the **log-likelihood function** for the given data where:

- \( X = 45 \) (feature)
- Label \( Y = 0 \) (actual outcome)
- Predicted probability \( \hat{p} = 0.95 \)

The log-likelihood function for binary logistic regression is given by:

\[
LL(\beta) = Y \cdot \log(\hat{p}) + (1 - Y) \cdot \log(1 - \hat{p})
\]

Substituting the given values:

- \( Y = 0 \)
- \( \hat{p} = 0.95 \)

\[
LL(\beta) = 0 \cdot \log(0.95) + (1 - 0) \cdot \log(1 - 0.95)
\]
\[
LL(\beta) = \log(0.05)
\]

Now, we calculate \( \log(0.05) \):

\[
LL(\beta) = \log(0.05) \approx -2.9957
\]

Thus, the log-likelihood value is approximately **-2.996**.

Let me know if you want further details or have any questions!

Here are five related questions to expand on this:
1. How is the log-likelihood function derived in logistic regression?
2. What is the interpretation of the log-likelihood value in logistic models?
3. How does the log-likelihood change with different predicted probabilities?
4. What role does the log-likelihood play in model evaluation?
5. How can you use log-likelihood in the context of maximum likelihood estimation (MLE)?

**Tip:** In logistic regression, maximizing the log-likelihood function helps in estimating the model's parameters that best fit the observed data.

Understand how to evaluate the log-likelihood function in binary logistic regression with data X=45, label Y=0, and predicted probability 0.95. This solution explores key concepts such as the log-loss function, log-likelihood, and maximum likelihood estimation. Suitable for undergraduate students.

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