Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Problem Statement:
In the diagram of triangle \( \triangle NRQ \) below, segment \( SP \parallel RQ \), \( NS = 3 \), \( SR = 21 \), and \( PQ = 63 \). What is the length of \( NQ \)?
### Diagram Description:
The diagram illustrates triangle \( \triangle NRQ \) with the following details:
- Point \( N \) is at the top vertex.
- Point \( R \) is at the left vertex.
- Point \( Q \) is at the right vertex.
- Point \( S \) is on segment \( NR \).
- Point \( P \) is on segment \( NQ \).
- Segment \( SP \) is parallel to segment \( RQ \), indicated by \( SP \parallel RQ \).
- Length \( NS = 3 \).
- Length \( SR = 21 \).
- Length \( PQ = 63 \).
### Solution Approach:
1. Identify the triangle \( \triangle NRQ \).
2. Note that \( SP \parallel RQ \) implies triangle similarity by the Basic Proportionality Theorem (also known as Thales' theorem).
3. Use the given measurements to determine the length \( NQ \).
### Calculation Steps:
1. Since \( SP \parallel RQ \), triangles \( \triangle NSP \) and \( \triangle NRQ \) are similar.
2. By the proportionality of similar triangles:
\[ \frac{NS}{NR} = \frac{NP}{NQ} \]
3. Calculate \( NR \):
\[ NR = NS + SR = 3 + 21 = 24 \]
4. Use the ratio to find \( NQ \):
\[ \frac{NS}{NR} = \frac{3}{24} = \frac{1}{8} \]
\[ \frac{NP}{NQ} = \frac{1}{8} \]
Given \( PQ = 63 \), we have:
\[ NQ = NP + PQ = 8 \times NP \]
\[ PQ = 63 \]
Hence, \( NP = PQ = 63 / (8 - 1) = 63 / 7 = 9 \)
Finally,
\[ NQ = 63 + 9 = 72 \]
### Answer Submission:
Answer: \( 72 \)
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