California Recidivism In California, the recidivism rate for prisoners is 67.5 % . That is 67.5 % of those released from prison go back to prison within three years. This is one of the highest recidivism rates in the nation. a. Suppose two independent prisoners are released. What is the probability that they will both go back to prison within three years? b. What is the probability that neither will go back to prison within three years? c. Suppose two independent prisoners are released. What is the probability that one OR the other (or both) will go back to prison within three years?
California Recidivism In California, the recidivism rate for prisoners is 67.5 % . That is 67.5 % of those released from prison go back to prison within three years. This is one of the highest recidivism rates in the nation. a. Suppose two independent prisoners are released. What is the probability that they will both go back to prison within three years? b. What is the probability that neither will go back to prison within three years? c. Suppose two independent prisoners are released. What is the probability that one OR the other (or both) will go back to prison within three years?
Solution Summary: The author calculates the probability that two randomly selected released prisoners will go back to prison within three years. The probability of observing an independent event must add up to 1.
California Recidivism In California, the recidivism rate for prisoners is
67.5
%
.
That is
67.5
%
of those released from prison go back to prison within three years. This is one of the highest recidivism rates in the nation.
a. Suppose two independent prisoners are released. What is the probability that they will both go back to prison within three years?
b. What is the probability that neither will go back to prison within three years?
c. Suppose two independent prisoners are released. What is the probability that one OR the other (or both) will go back to prison within three years?
3. Bayesian Inference – Updating Beliefs
A medical test for a rare disease has the following characteristics:
Sensitivity (true positive rate): 99%
Specificity (true negative rate): 98%
The disease occurs in 0.5% of the population.
A patient receives a positive test result.
Questions:
a) Define the relevant events and use Bayes’ Theorem to compute the probability that the patient actually has the disease.b) Explain why the result might seem counterintuitive, despite the high sensitivity and specificity.c) Discuss how prior probabilities influence posterior beliefs in Bayesian inference.d) Suppose a second, independent test with the same accuracy is conducted and is also positive. Update the probability that the patient has the disease.
4. Linear Regression - Model Assumptions and Interpretation
A real estate analyst is studying how house prices (Y) are related to house size in square feet (X). A simple
linear regression model is proposed:
The analyst fits the model and obtains:
•
Ŷ50,000+150X
YBoB₁X + €
•
R² = 0.76
• Residuals show a fan-shaped pattern when plotted against fitted values.
Questions:
a) Interpret the slope coefficient in context.
b) Explain what the R² value tells us about the model's performance.
c) Based on the residual pattern, what regression assumption is likely violated? What might be the
consequence?
d) Suggest at least two remedies to improve the model, based on the residual analysis.
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Discrete Distributions: Binomial, Poisson and Hypergeometric | Statistics for Data Science; Author: Dr. Bharatendra Rai;https://www.youtube.com/watch?v=lHhyy4JMigg;License: Standard Youtube License