목차

• Continuous Probability Distributions
• The Normal Distribution
• The Standardized Normal
• Finding Normal Probabilities
• 문제

👀, 🤷‍♀️ , 📜, 📝
이 아이콘들을 누르시면 정답, 개념 부가 설명을 보실 수 있습니다:)

Continuous Probability Distributions

[continuous variable]
a variable that can assume any value on a continuum (can assume an uncountable number of values)
• thickness of an item
• time required to complete a task
• temperature of a solution
• height, in inches

These can potentially take on any value depending only on the ability to precisely and accurately measure

The Normal Distribution

정규 분포

• ‘Bell Shaped’: 가운데가 볼록해야한다(필수조건)
• Symmetrical: 대칭이여야 한다(필수조건)
• Mean, Median and Mode are Equal

[plot]

• Location is determined by the mean, μ
• Spread is determined by the standard deviation, σ
• The random variable has an infinite theoretical range: + ∞ to - ∞

[formula]

• e = the mathematical constant approximated by 2.71828
• π = the mathematical constant approximated by 3.14159
• μ = the population mean
• σ = the population standard deviation
• X = any value of the continuous variable

[Distribution]

• A and B have the same mean but different standard deviations.
• B and C have different means and different standard deviations.

The Standardized Normal

표준정규분포
= 평균이 0이고 분산이 1인 정규분포

• Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normaldistribution (Z)
  ➡️ 모든 정규분포(평균 및 표준편차 조합 포함)는 표준화된 정규분포(Z)로 변환 가능
• To compute normal probabilities need to transform X units into Z units
  ➡️ 정규 확률을 계산하려면 X 단위를 Z 단위로 변환 필요
• The standardized normal distribution (Z) has a mean of 0 and a standard deviation of 1

[formula of Translation]
Translate from X to the standardized normal (the “Z” distribution) by subtracting the mean of X and dividing by its standard deviation

The Z distribution always has,

• mean = 0
• standard deviation = 1

[formula]

• e = the mathematical constant approximated by 2.71828
• π = the mathematical constant approximated by 3.14159
• Z = any value of the standardized normal distributio

[Distribution]
• Also known as the “Z” distribution
• Mean is 0
• Standard Deviation is 1

• Values above the mean have positive Z-values.
• Values below the mean have negative Z-value

❓ Comparing X and Z units

Note that the shape of,

• the distribution is the same
• only the scale has changed

We can express the problem in the original units (X in dollars) or in standardized units (Z)

Finding Normal Probabilities

Probability is measured by the area under the curve
이제 정규분포에서 아래와 같이 연속적인 확률의 합을 구하는 방법을 볼 것이다.

[Probability as Area Under the Curve]
The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

[The Standardized Normal Table]
또한 계산 결과 z 값의 확률은 The Standardized Normal Table로 판별 가능하다

💡 STEP TO FIND Normal Probabilities 💡
1)

• Suppose X is normal with a mean of18.0 seconds and a standard deviation of 5.0 seconds.
• Find P(X < 18.6)
  

2)

• Let X represent the time it takes, in seconds to download an image file from the internet.
• Suppose X is normal with a mean of 18.0 seconds and a standard deviation of 5.0 seconds. Find P(X < 18.6)
  

3)

• Standardized Normal Probability Table (Portion)
  

📌 Finding Normal Upper Tail Probabilities

문제

[Table 6-1] You were told that the amount of time lapsed between consecutive trades on the New York Stock Exchange followed a normal distribution with a mean of 15 seconds. You were also told that the probability that the time lapsed between two consecutive trades to fall between 16 to 17 seconds was 13%. The probability that the time lapsed between two consecutive trades would fall below 13 seconds was 7%.

1. Referring to Table 6-1, what is the probability that the time lapsed between two consecutive trades will be between 14 and 15 seconds?

2. Referring to Table 6-1, what is the probability that the time lapsed between two consecutive trades will be between 13 and 16 seconds?

3. Referring to Table 6-1, what is the probability that the time lapsed between two consecutive trades will be between 14 and 17 seconds?

4. If a particular batch of data is approximately normally distributed, we would find that approximately
a) 2 of every 3 observations would fall between 1 standard deviation around the mean.
b) 4 of every 5 observations would fall between 1.28 standard deviations around the mean.
c) 19 of every 20 observations would fall between 2 standard deviations around the mean.
d) All the above.

5. For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3770. The value of Z is

6. For some value of Z, the probability that a standard normal variable is below Z is 0.2090. The value of Z is

7. The probability that a standard normal variable Z is positive is ________.

8. A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation is 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall below 10.875 ounces.

9. A food processor packages orange juice in small jars. The weights of the filled jars are approximately normally distributed with a mean of 10.5 ounces and a standard deviation is 0.3 ounce. Find the proportion of all jars packaged by this process that have weights that fall above 10.95 ounces.