Statistical Inference Solutions: Estimation and Hypothesis Testing
Q1 True/False (each needs a quick justification)
Q1(a) Binomial(100, 0.2) approximated by Normal N(20,16)
Steps:
Identify: X ~ Bin(n=100, p=0.2).
Compute mean: μ = np = 100(0.2) = 20.
Compute variance: σ² = np(1−p) = 100(0.2)(0.8) = 16.
Rule: For large n, Binomial ≈ Normal with same mean/variance (CLT-ish approximation).
Conclusion: Yes, approximate with N(20,16).
Q1(b) MAP estimator equals argmax log posterior
Steps:
MAP definition: θ̂MAP = argmaxθ f(θ|x).
Taking log does not move the maximizer
Essential Templates for Data Visualization Reporting
Line Graph Analysis Template
- The line graph illustrates trends in [subject] over a [time period] from [start date] to [end date]. The data is measured in [units] and demonstrates [general observation].
- Overall, the most significant trend is the [adjective] [increase/decrease] in [main subject]. While [item A] showed substantial growth, [item B] experienced considerable fluctuation. The period witnessed [item C] consistently [outperforming/underperforming] other categories.
- Beginning with [item A],
Geometry Fundamentals and Logic Principles
Chapter 1: Foundations of Geometry
1.1: Points, Lines, and Planes
- Collinear: Points on the same line.
- Coplanar: Points that lie in the same plane.
- Segment: A part of a line consisting of two endpoints and all points between them.
- Endpoint: A point at the end of a segment or the starting point of a ray.
- Ray: Part of a line that extends from one endpoint infinitely in one direction.
- Opposite rays: Two rays that have a common endpoint and form a line.
- Postulate: A statement that is accepted as true without
Engineering Investment Evaluation: Present, Future & Annual Worth
Present, Future & Annual Equivalent — Equal & Unequal Lives
1. Purpose of Investment Evaluation
Used to compare different engineering projects based on their economic worth.
Projects may have:
- Different lifetimes
- Different cash-flow patterns
- Different initial investments
Goal → Select the economically best alternative.
2. Interest Factors (Essential)
(These appear multiple times, so memorize.)
Let i = interest rate per period, n = number of periods.
(P/F, i, n)
Present worth of single future sum:
Geospatial Data Models and Coordinate Systems Explained
Geospatial Data Models and Acquisition
The Spaghetti Data Model lacks essential spatial relationships such as connectivity (chaining), adjacency (left/right), or containment (e.g., Polygon X within Polygon Y).
Elements of GIScience: Geospatial Data
Geospatial analysis relies on two primary data types:
- Vector Data: Represents discrete objects with distinct boundaries.
- Raster Data: Represents continuous fields where every location has a value.
Data Acquisition Methods
Data acquisition involves obtaining
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JULIOL 2018 B 1. A)par 2: companies Whose… B) par 3: currently, there… 2. A)F 250 bottles… B)F “We found… C)F people in developing… 3.Concerning, Investigation, meet, pervasive 4. A,b,b.
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