Chapter 6: Continuous Probability Distribution

The probability that the normal random variable Z is less than 1.5 is equal to...

P(Z > -1.5)

For a discrete random variable X,

there are a countable number of possible values.

The probability distribution of a continuous random variable is called its probability...

density function.

For a continuous random variable, one characteristic of its probability density function f(x) is that the area under f(x) over all possible values of x is....

equal to one. - total probability always equals 100% = 1.0.

A continuous random variable X follows the uniform distribution with a lower limit of a and an upper limit of b. The expected value of X is calculated as ____________.

(a + b) / 2

How many parameters are needed to fully describe any normal distribution?

- 2 > Two parameters are needed, the mean and the standard deviation.

Which of the following random variables is depicted with a bell-shaped curve? - An exponential random variable - A uniform random variable - A normal random variable - A binomial random variable

- A normal random variable

The mean and variance of the standard normal distribution are ____________, respectively.

0 and 1.

The variance of the standard normal distribution is equal to ________.

1. 1 is also the standard deviation.

Which of the following continuous distributions are positively skewed and bounded below by zero? a) The lognormal distribution b) The uniform distribution c) The exponential distribution d) The normal distribution

a) The lognormal distribution c) The exponential distribution

If X is a normally distributed random variable, then...

the mean, the median, and the mode are all equal.

Continuous

the random variable is characterized by (infinitely) uncountable values within any interval.

Consider data that are normally distributed. In order to transform a standard deviation normal value z into its unstandardized value x, we use the following formula:

x = population mean + (z * population standard deviation)

The probability that a continuous random variable X assumes a particular values x is:

zero.

Probability Density Function f(x) of a continuous random variable X.

- Describes the relative likelihood that X assumes a values within a given interval, where: > f(x) > 0 for all possible values of X. > The area under f(x) over all values of x equals one.

A continuous random variable has the uniform distribution of the interval [a,b] if its probability density function f(x):

is constant for all x between a and b, and 0 otherwise.

Suppose you were told that the delivery time of your new washing machine is equally likely over the time period 9 am to noon. If we define the random variable X as delivery time, then X follows the...

continuous uniform distribution.

For a continuous random variable, one characteristic of its probability density function f(x) is that....

f(x) is greater than or equal to 0 for all values x of X.

For a continuous random variable X it is only meaningful to calculate the probability that the value of the random variable:

falls within some specified interval.

A characteristic of the normal distribution is that...

it is symmetric around its mean.

The normal distribution is completely described by these 2 parameters:

mean and variance

Discrete

the random variable assumes a countable number of distinct values.

due to symmetry, the probability that the normal random variable Z is greater than 1.5 is equal to

P(Z< -1.5)

The probability that a discrete random variable X assumes a particular value x is...

between 0 and 1.

Which of the following is an example of a continuous random variable? a) A Bernoulli trial b) Binomial random variable c) Normal random variable d) Discrete uniform random variable

c) Normal random variable

A random variable X with an equally likely chance of assuming any value within a specified range is said to have which distribution?

continuous uniform distribution

Suppose you were told that the delivery time of your new washing machine is equally likely over the time period 9 am to noon. If we define the random variable X as delivery time, the X follows the..

continuous uniform distribution.

The mean of the standard normal distribution is equal to ____________.

zero - That's the center of the Z curve.

Continuous Uniform Distribution

- Describes a random variable that has an equally likely chance of assuming a value within a specified range.

Due to symmetry, the probability that the standard normal random variable Z is less than 0 is equal to ________.

0.5 - The left half of the Z curve.

The probability that a normal random variable X is less than its mean is equal to...

0.50 (since the bell curve shows perfect symmetry between the left and the right sides)

For a continuous random variable X, the function used to find the are under f(x) up to any value x is called the...

cumulative distribution function.

A manager of a women's clothing store is projecting next month's sales. Her low-end estimate of sales is $25,000 and her high-end estimate is $50,000. She decides to treat all outcomes for sales between these two values as equally likely. If we define the random variable X as sales, then X follows the...

uniform distribution.

The height of the probability density function f(x) of the uniform distribution defined on the interval [a,b] is...

1 / (b - a) between a and b, and zero otherwise.

The inverse transformation, x= population mean + (z*population standard deviation) is used to ____________.

Compute x values for given probabilities.

It is known that the length of a certain product X is normally distributed with population mean = 20 inches. How is the P(X < 20) related to the P(X < 16)?

P(X < 20) is greater than P(X < 16).

Continuous Random Variables and the Uniform Distribution

When computing probabilities for a continuous random variable, keep in mind that P(X=x) = 0. - We cannot assign a nonzero probability to each infinitely uncountable value and still have the probabilities sum to one. - Thus, since P(X= a) and P(X= b) both equal zero, the following holds for continuous random variables: (refer to the attached image)

Which of the following can be represented by a continuous random variable? a) The high daily temperature in Tampa, Florida during the month of July, measured in degrees. b) The number of customers who visit a department store between 10:00 am and 11:00 am on Mondays. c) The number of typos found on a randomly selected page of a textbook. d) The number of students who will get financial assistance in a group of 50 randomly selected students.

a) The high daily temperature in Tampa, Florida during the month of July, measured in degrees.

Cumulative Density Function F(x) of a continuous random variable X.

- For any value x of the random variable X, the cumulative distribution function F(x) is computed as : (refer to the attached image)

All of the following are examples of random variables that likely follow a normal distribution EXCEPT: - The scores on the SAT exam - Income in the USA - The debt of college graduates - The weights of newborn babies

- Income in the USA ---In general, income in the USA is positively skewed.

A characteristic of the normal distribution is...

- It is asymptotic.

Normal Distribution

- Symmetric - Bell-Shaped - Closely approximates the probability distribution of a wide range of random variables such as, Heights and Weights of newborn babies, Scores on SAT, Cumulative debt of college graduates. - Serves as the cornerstone of statistical inference.

Characteristics of the Normal Distribution

- Symmetric about its mean. > Mean = Median = Mode - Asymptotic - that is, the tails get closer and closer to the horizontal axis, but never touch it. - The normal distribution is completely described by two parameters.

Graph of the continuous uniform distribution:

- The values a and b on the horizontal axis represent the lower and upper limits, respectively. - The height of the distribution does not directly represent a probability. - It is the are under f(x) that corresponds to probability.

The total area under the normal v=curve is _______.

- equal to 1 > Total probability equals 100% or 1.0.

For a continuous random variable X it is only meaningful to calculate the probability that the value of the random variable....

- falls within some specified interval.

The z table provides the cumulative probability for a given value z. What does "cumulative probability" mean?

The probability that Z is less than or equal to a given z value.

Consider data that are normally distributed. In order to transform a value x into it standardized value z, we use the following formula:

z = x - population mean / population standard deviation