2. Some Continuous Probability Distributions
Chapter 6
Some Continuous Probability Distributions
Chapter 6
Some Continuous Probability Distributions

3. Continuous Uniform Distribution
Chapter 6.1
Continuous Uniform Distribution
Continuous Uniform Distribution
|Uniform Distribution| The density function of the continuous uniform random variable X on the interval [A, B] is
The mean and variance of the uniform distribution are
and
The uniform density function for a random variable on the interval [1, 3]

4. Continuous Uniform Distribution
Chapter 6.1
Continuous Uniform Distribution
Continuous Uniform Distribution
Suppose that a large conference room for a certain company can be reserved for no more than 4 hours. However, the use of the conference room is such that both long and short conference occur quite often. In fact, it can be assumed that length X of a conference has a uniform distribution on the interval [0, 4].
What is the probability density function?
What is the probability that any given conference lasts at least 3 hours?

5. Chapter 6.2
Normal Distribution
Normal Distribution
Normal distribution is the most important continuous probability distribution in the entire field of statistics.
Its graph, called the normal curve, is the bell-shaped curve which describes approximately many phenomena that occur in nature, industry, and research.
The normal distribution is often referred to as the Gaussian distribution, in honor of Karl Friedrich Gauss, who also derived its equation from a study of errors in repeated measurements of the same quantity.
The normal curve

6. Chapter 6.2
Normal Distribution
Normal Distribution
A continuous random variable X having the bell-shaped distribution as shown on the figure is called a normal random variable.
The density function of the normal random variable X, with mean μ and variance σ2, is
where π = and e =

7. Normal Curve μ1 < μ2, σ1 = σ2 μ1 = μ2, σ1 < σ2
Chapter 6.2
Normal Distribution
Normal Curve
μ1 < μ2, σ1 = σ2
μ1 = μ2, σ1 < σ2
μ1 < μ2, σ1 < σ2

8. Chapter 6.2
Normal Distribution
Normal Curve x
f(x) μ
The mode, the point where the curve is at maximum
Concave downward
Point of inflection σ
Concave upward
Approaches zero asymptotically
Total area under the curve and above the horizontal axis is equal to 1
Symmetry about a vertical axis through the mean μ

9. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
The area under the curve bounded by two ordinates x = x1 and x = x2 equals the probability that the random variable X assumes a value between x = x1 and x = x2.

10. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
As seen previously, the normal curve is dependent on the mean μ and the standard deviation σ of the distribution under investigation.
The same interval of a random variable can deliver different probability if μ or σ are different.
Same interval, but different probabilities for two different normal curves

11. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
The difficulty encountered in solving integrals of normal density functions necessitates the tabulation of normal curve area for quick reference.
Fortunately, we are able to transform all the observations of any normal random variable X to a new set of observation of a normal random variable Z with mean 0 and variance 1.

12. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution.

13. Table A.3 Normal Probability Table
Chapter 6.3
Areas Under the Normal Curve
Table A.3 Normal Probability Table

14. Chapter 6.3
Areas Under the Normal Curve
Interpolation
Interpolation is a method of constructing new data points within the range of a discrete set of known data points.
Examine the following graph. Two data points are known, which are (a,f(a)) and (b,f(b)).
If a value of c is given, with a < c < b, then the value of f(c) can be estimated.
If a value of f(c) is given, with f(a) < f(c) < f(b), then the value of c can be estimated.

15. Interpolation P(Z < 1.172)? Answer: 0.8794
Chapter 6.3
Areas Under the Normal Curve
Interpolation
P(Z < 1.172)? Answer:
P(Z < z) = , z = ?

16. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given a standard normal distribution, find the area under the curve that lies (a) to the right of z = 1.84 and (b) between z = –1.97 and z = 0.86.

17. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given a standard normal distribution, find the value of k such that (a) P ( Z > k ) = , and (b) P ( k < Z < –0.18 ) =

18. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given a random variable X having a normal distribution with μ = 50 and σ = 10, find the probability that X assumes a value between 45 and 62.

19. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given that X has a normal distribution with μ = 300 and σ = 50, find the probability that X assumes a value greater than 362.

20. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given a normal distribution with μ = 40 and σ = 6, find the value of x that has (a) 45% of the area to the left, and (b) 14% of the area to the right.

21. Area Under the Normal Curve
Chapter 6.3
Areas Under the Normal Curve
Area Under the Normal Curve
Given a normal distribution with μ = 40 and σ = 6, find the value of x that has (a) 45% of the area to the left, and (b) 14% of the area to the right.

22. Applications of the Normal Distribution
Chapter 6.4
Applications of the Normal Distribution
Applications of the Normal Distribution
A certain type of storage battery lasts, on average, 3.0 years, with a standard deviation of 0.5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last less than 2.3 years.

23. Applications of the Normal Distribution
Chapter 6.4
Applications of the Normal Distribution
Applications of the Normal Distribution
In an industrial process the diameter of a ball bearing is an important component part. The buyer sets specifications on the diameter to be 3.0 ± 0.01 cm. All parts falling outside these specifications will be rejected.
It is known that in the process the diameter of a ball bearing has a normal distribution with mean 3.0 and standard deviation
On the average, how many manufactured ball bearings will be scrapped?

24. Applications of the Normal Distribution
Chapter 6.4
Applications of the Normal Distribution
Applications of the Normal Distribution
A certain machine makes electrical resistors having a mean resistance of 40 Ω and a standard deviation of 2 Ω. It is assumed that the resistance follows a normal distribution.
What percentage of resistors will have a resistance exceeding 43 Ω if:
the resistance can be measured to any degree of accuracy.
the resistance can be measured to the nearest ohm only.
As many as 6.68%–4.01% = 2.67% of the resistors will be accepted although the value is greater than 43 Ω due to measurement limitation

25. Applications of the Normal Distribution
Chapter 6.4
Applications of the Normal Distribution
Applications of the Normal Distribution
The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class are given A’s, and the grade are curved to follow a normal distribution, what is the lowest possible A and the highest possible B?
Lowest possible A is 83
Highest possible B is 82

26. Probability and Statistics
Homework 7A
Suppose the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 milliamperes2. (a) What is the probability that a measurement will exceed 13 milliamperes? (b) Determine the value for which the probability that a current measurement is below this value is 98%. (Mo.E p.113)
A lawyer commutes daily from his suburban home to midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed. (a) If the office opens at 9:00 A.M. and the lawyer leaves his house at 8:45 A.M. daily, what percentage of the time is he late for work? (b) Find the probability that 2 of the next 3 trips will take at least 1/2 hour. (Wa.6.15 s.186)