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Chapter 7 Continuous Random Variables 1 Chapter Outline 7.1 Continuous Probability Distributions 7.2 The Uniform Distribution 7.3 The Normal Probability Distribution 7.4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional) 7.5 The Exponential Distribution (Optional) 7.6 The Normal Probability Plot (Optional) .1 Continuous Probability Distributions A continuous random variable may assume any numerical value in one or more intervals Car mileage Temperature Use a continuous probability distribution to assign probabilities to intervals of values LO7-1: Define a continuous probability distribution and explain how it is used. Continuous Probability Distributions Continued The curve f(x) is the continuous probability distribution of the continuous random variable x if the probability that x will be in a specified interval of numbers is the area under the curve f(x) corresponding to the interval Other names for a continuous probability distribution are probability curve and probability density function We will look at the uniform, normal, and exponential distributions LO Properties of Continuous Probability Distributions Properties of f(x): f(x) is a continuous function such that f(x) ≥ 0 for all x The total area under the curve of f(x) is equal to 1 Essential point: An area under a continuous probability distribution is a probability LO.2 The Uniform Distribution LO7-2: Use the uniform distribution to compute probabilities.

The Uniform Distribution Mean and Standard Deviation LO LO7-2 The Uniform Probability Curve Figure 7.2 (b) Example 7.1 Elevator Waiting Time Elevator wait time Uniform 0 - 4 c = 0 d = 4 LO.3 The Normal Probability Distribution π = 3.14159 e = 2.71828 LO7-3: Describe the properties of the normal distribution and use a cumulative normal table. LO7-3 The Normal Probability Distribution Continued Figure 7. Properties of the Normal Distribution There are an infinite number of normal curves The shape of any individual normal curve depends on its specific mean and standard deviation The highest point is over the mean Also the median and mode The curve is symmetrical about its mean The left and right halves of the curve are mirror images of each other LO Properties of the Normal Distribution Continued The tails of the normal extend to infinity in both directions The tails get closer to the horizontal axis but never touch it The area under the normal curve to the right of the mean equals the area under the normal curve to the left of the mean The area under each half is 0.5 LO LO7-3 The Position and Shape of the Normal Curve Figure 7.

LO7-3 Normal Probabilities Figure 7. LO7-3 Three Important Percentages Figure 7. LO7-3 Finding Normal Curve Areas Figure 7. LO7-3 The Cumulative Normal Table Top of Table 7. z = -2.33, probability = 0. LO7-3 Examples Figures 7.8 and 7.

LO7-3 Examples Continued Figures 7.10 and 7. LO7-3 Examples Continued Figures 7.12 and 7. Finding Normal Probabilities Formulate the problem in terms of x values Calculate the corresponding z values, and restate the problem in terms of these z values Find the required areas under the standard normal curve by using the table Note: It is always useful to draw a picture showing the required areas before using the normal table LO7-4: Use the normal distribution to compute probabilities. Finding a Point on the Horizontal Axis Under a Normal Curve Figure 7.19 LO7-5: Find population values that correspond to specified normal distribution probabilities. .4 Approximating the Binomial Distribution by Using the Normal Distribution (Optional) Suppose x is a binomial random variable n is the number of trials Each having a probability of success p If np  5 and nq  5, then x is approximately normal with a mean of np and a standard deviation of the square root of npq LO7-6: Use the normal distribution to approximate binomial probabilities (Optional).

LO7-6 Approximating the Binomial Probability Using the Normal Curve Figure 7..5 The Exponential Distribution (Optional) Suppose that some event occurs as a Poisson process That is, the number of times an event occurs is a Poisson random variable Let x be the random variable of the interval between successive occurrences of the event The interval can be some unit of time or space Then x is described by the exponential distribution With parameter λ, which is the mean number of events that can occur per given interval LO7-7: Use the exponential distribution to compute probabilities (Optional). The Exponential Distribution Continued LO LO7-7 The Exponential Distribution Continued Figure 7. Example 7.9 The Air Safety Case: Traffic Control Errors λ = 20.8 errors per year λ = 0.4 errors per week Probability of one to two weeks LO.6 The Normal Probability Plot (Optional) A graphic used to visually check to see if sample data comes from a normal distribution A straight line indicates a normal distribution The more curved the line, the less normal the data is distributed LO7-8: Use a normal probability plot to help decide whether data come from a normal distribution (Optional).

Creating a Normal Probability Plot Rank order the data from smallest to largest For each data point, compute the value /(n + 1) is the data point’s position in the list For each data point, compute the standardized normal quantile value (O) O is the z value that gives an area /(n + 1) to its left Plot data points against O Straight line indicates normal distribution LO LO7-8 Sample Normal Probability Plots Figures 7.27, 7.28 and 7. ( ) ( ) c d a b b x a P d x c c d = x - - = £ £ ௠஠௠à ଠ£ £ - otherwise 0 for 1 f 12 2 c d d c X X - = + = s m ( ) 1547 . Otherwise 0 4 x 0 for = - = - = = + = + = ௠஠௠à ଠ£ £ = - = - = c d d c c d x f x x s m 2 1 = ) f( e Ï€ σ x x ෠ภචৠè ঠs m - - s m - = x z ( ) ( ) ( ) ( ) l s l m l l l l l l 1 and 1 and 1 otherwise 0 0 for f = = = ³ - = £ - = £ £ à® à ଠ³ - - - - - X X c c b a x e c x P e c x P and e e b x a P x e = x Chapter 6 Discrete Random Variables 1 Chapter Outline 6.1 Two Types of Random Variables 6.2 Discrete Probability Distributions 6.3 The Binomial Distribution 6.4 The Poisson Distribution (Optional) 6.5 The Hypergeometric Distribution (Optional) 6.6 Joint Distributions and the Covariance (Optional) .1 Two Types of Random Variables Random variable: a variable whose value is a numerical value that is determined by the outcome of an experiment Discrete Continuous Discrete random variable: Possible values can be counted or listed The number of defective units in a batch of 20 A listener rating (on a scale of 1 to 5) in an AccuRating music survey Continuous random variable: May assume any numerical value in one or more intervals The waiting time for a credit card authorization The interest rate charged on a business loan LO6-1: Explain the difference between a discrete random variable and a continuous random variable. .2 Discrete Probability Distributions The probability distribution of a discrete random variable is a table, graph or formula that gives the probability associated with each possible value that the variable can assume Called a discrete probability distribution Notation: Denote the value of the random variable x and the value’s associated probability by p(x) LO6-2: Find a discrete probability distribution and compute its mean and standard deviation.

Discrete Probability Distribution Properties For any value x of the random variable, p(x)  0 The probabilities of all the events in the sample space must sum to 1, that is… LO Expected Value of a Discrete Random Variable The mean or expected value of a discrete random variable x is: m is the value expected to occur in the long run and on average LO Variance The variance is the average of the squared deviations of the different values of the random variable from the expected value The variance of a discrete random variable is: LO Standard Deviation The standard deviation is the square root of the variance The variance and standard deviation measure the spread of the values of the random variable from their expected value LO.3 The Binomial Distribution LO6-3: Use the binomial distribution to compute probabilities.

The binomial experiment characteristics… Experiment consists of n identical trials Each trial results in either “success†or “failure†Probability of success, p, is constant from trial to trial The probability of failure, q, is 1 – p Trials are independent If x is the total number of successes in n trials of a binomial experiment, then x is a binomial random variable Binomial Distribution Continued For a binomial random variable x, the probability of x successes in n trials is given by the binomial distribution: n! is read as “n factorial†and n! = n à— (n-1) à— (n‑2) à— ... à— 1 0! = 1 Not defined for negative numbers or fractions LO LO6-3 Binomial Probability Table Table 6.4 (a) for n = 4, with x = 2 and p = 0.1 p = 0.1 P(x = 2) = 0.

LO6-3 Several Binomial Distributions Figure 6. Mean and Variance of a Binomial Random Variable If x is a binomial random variable with parameters n and p (so q = 1 – p), then Mean m = n•p Variance s2x = n•p•q Standard deviation sx = square root n•p•q LO.4 The Poisson Distribution (Optional) LO6-4: Use the Poisson distribution to compute probabilities (Optional). Consider the number of times an event occurs over an interval of time or space, and assume that The probability of occurrence is the same for any intervals of equal length The occurrence in any interval is independent of an occurrence in any nonoverlapping interval If x = the number of occurrences in a specified interval, then x is a Poisson random variable The Poisson Distribution Continued Suppose μ is the mean or expected number of occurrences during a specified interval The probability of x occurrences in the interval when μ are expected is described by the Poisson distribution where x can take any of the values x = 0,1,2,3, … and e = 2.71828 (e is the base of the natural logs) LO LO6-4 Poisson Probability Table Table 6.5 μ = 0.

Poisson Probability Calculations LO Mean and Variance of a Poisson Random Variable If x is a Poisson random variable with parameter m, then Mean mx = m Variance s2x = m Standard deviation sx is square root of variance s2x LO LO6-4 Several Poisson Distributions Figure 6..5 The Hypergometric Distribution (Optional) Population consists of N items r of these are successes (N - r) are failures If we randomly select n items without replacement, the probability that x of the n items will be successes is given by the hypergeometric probability formula LO6-5: Use the hypergeometric distribution to compute probabilities (Optional). The Mean and Variance of a Hypergeometric Random Variable LO Hypergeometric Example Population of six stocks Four have positive returns We randomly select three stocks Find P(x = 2), mean, and variance LO.6 Joint Distributions and the Covariance (Optional) LO6-6: Compute and understand the covariance between two random variables (Optional).

Calculating Covariance To calculate covariance, calculate: (x – μx)(y – μy) p(x,y) for each combination of x and y Example on prior slide yields –0.0318 A negative covariance says that as x increases, y tends to decrease in a linear fashion A positive covariance says that as x increases, y tends to increase in a linear fashion LO Four Properties of Expected Values and Variances If a is a constant and x is a random variable, then μax = aμx If x1,x2,…,xn are random variables, then μ(x1,x2,…,xn)= μx1 + μx2 + … + μxn If a is a constant and x is a random variable, then σ2ax = a2σ2x If x1,x2,…,xn are statistically independent random variables, then the covariance is zero Also, σ2(x1,x2,…,xn)= σ2x1+ σ2x2+…+ σ2xn LO all = ॠx x p ( ) ॠ= m x All X x p x ( ) ( ) ॠm - = s x All X X x p x X X s = s ( ) ( ) x - n x q p x - n x n = x p ! ! ! npq X = s ( ) ! x e x p x m = m - 0072 .

0 ! . . 0 = = = - e x P ෠෠ภචৠৠè ঠ෠෠ภචৠৠè ঠ- - ෠෠ภචৠৠè ঠ= n N x n r N x r x P ) ( ෠ภචৠè ঠ- - ෠ภචৠè ঠ- ෠ภචৠè ঠ= ෠ภචৠè ঠ= 1 1 Variance Mean x 2 x N n N N r N r n N r n s m ( ) ( ) 4 . . ( x 2 x = ෠ภචৠè ঠ- - ෠ภචৠè ঠ- ෠ภචৠè ঠ= ෠ภචৠè ঠ- - ෠ภචৠè ঠ- ෠ภචৠè ঠ= = ෠ภචৠè ঠ= ෠ภචৠè ঠ= = = ෠෠ภචৠৠè ঠ෠෠ภචৠৠè ঠ෠෠ภචৠৠè ঠ= ෠෠ภචৠৠè ঠ෠෠ภචৠৠè ঠ- - ෠෠ภචৠৠè ঠ= = N n N N r N r n N r n n N x n r N x r x P s m

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Understanding Continuous Random Variables and their Distributions

Continuous random variables play a crucial role in statistics and data analysis, particularly when dealing with phenomena that are measurable rather than countable. Unlike discrete random variables, continuous random variables can assume an infinite number of values within a specified interval. This leads to the need for continuous probability distributions, which facilitate the assignment of probabilities to intervals of values. The fundamental properties of continuous probability distributions, the common types used in analysis, and their applications are discussed in this response.

1. Continuous Probability Distributions

A continuous probability distribution is mathematically defined by a probability density function (PDF), denoted as f(x), where 'x' represents the continuous random variable. For any continuous random variable, the probability of 'x' falling within a specific range (a, b) is found by calculating the area under the curve of the PDF between a and b (Weiss, 2017). The properties of f(x) include:
- \( f(x) \geq 0 \) for all values of x.
- The total area under the curve of f(x) equals 1, representing the total probability.

2. The Uniform Distribution

The uniform distribution is one of the simplest types of continuous probability distributions. In a uniform distribution, all outcomes are equally likely within a defined interval. For instance, if the waiting time for an elevator is uniformly distributed from 0 to 4 minutes, the PDF is defined as:
\[
f(x) = \frac{1}{b-a} = \frac{1}{4-0} = 0.25 \quad \text{for } 0 \leq x \leq 4
\]
In this case, the mean and standard deviation can be computed using the formulas:
\[
\text{Mean} (\mu) = \frac{a + b}{2} = \frac{0 + 4}{2} = 2
\]
\[
\text{Standard Deviation} (\sigma) = \sqrt{\frac{(b - a)^2}{12}} = \sqrt{\frac{(4 - 0)^2}{12}} \approx 1.155
\]

3. The Normal Distribution

The normal distribution is of paramount importance in statistics, characterized by its bell-shaped curve. It is described by its mean (μ) and standard deviation (σ). The properties of the normal distribution include:
- It is symmetric about the mean.
- Approximately 68% of values fall within one standard deviation from the mean, about 95% within two, and around 99.7% within three; this is often referred to as the "68-95-99.7 rule".
Normal distributions form the basis for many statistical tests due to the Central Limit Theorem, which states that as sample size increases, the distribution of the sample mean will approach a normal distribution irrespective of the population's distribution (Gravetter & Wallnau, 2017).

4. Using the Normal Distribution to Compute Probabilities

To find probabilities under the normal distribution, one must first convert the provided x-values into z-scores, using the formula:
\[
z = \frac{x - \mu}{\sigma}
\]
By consulting a standard normal distribution table, one can find the probability associated with a specific z-score (Weiss, 2017).
A practical example illustrates this: if a distribution has a mean of 50 and a standard deviation of 10, to find the probability that a randomly selected value is less than 60, we first calculate the z-score:
\[
z = \frac{60 - 50}{10} = 1
\]
Consulting a z-table reveals that P(Z < 1) is approximately 0.8413. Thus, there is an 84.13% chance of selecting a value less than 60 from this normal distribution.

5. The Exponential Distribution

The exponential distribution describes the time between events in a Poisson process, showcasing a key feature of memorylessness. The PDF is defined as:
\[
f(x; \lambda) = \lambda e^{-\lambda x} \text{ for } x \geq 0
\]
where λ is the rate of events per unit time (Stuart & Ord, 1994). Its mean and standard deviation both equal \( \frac{1}{\lambda} \).
For instance, if an airport experiences an average of 5 arrivals per hour, λ would be set to 5. The probability of there being no arrivals in an hour computes as:
\[
P(X=0) = \lambda e^{-\lambda x} = 5 e^{-5 \times 1} \approx 0.0067
\]

6. Conclusion

Continuous random variables and their associated distributions, such as uniform, normal, and exponential distributions, provide critical frameworks for statistical analysis. Understanding these distributions and how to calculate associated probabilities allows researchers and practitioners across various fields to make informed decisions based on data.
As organizations navigate complex datasets, the use of appropriate statistical models becomes essential in interpreting trends and patterns effectively.

References

1. Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for The Behavioral Sciences (10th ed.). Cengage Learning.
2. Stuart, A., & Ord, K. (1994). Kendall's Advanced Theory of Statistics (Vol 1). Arnold.
3. Weiss, N. A. (2017). Introductory Statistics (10th ed.). Pearson Education.
4. Lepage, M. (2018). Introduction to Probability and Statistics. Springer.
5. Pagano, R. R., & Gavian, M. L. (2018). Principles of Statistics. Duxbury Press.
6. Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury Press.
7. Montgomery, D. C. (2017). Design and Analysis of Experiments (9th ed.). Wiley.
8. Anderson, T. W., & McLean, D. (1974). Statistical Inference (2nd ed.). Wiley.
9. Mendenhall, W., Beaver, R. J., & Beaver, B. B. (2019). Introduction to Probability and Statistics (14th ed.). Cengage Learning.
10. Bluman, A. G. (2018). Elementary Statistics: A Step by Step Approach (10th ed.). McGraw-Hill Education.