QM 214 6.1 & 6.2

Normal (probability) distribution-

The most extensively used probability distribution in statistical work and the cornerstone of statistical inference. It is symmetric and bell-shaped and is completely described by the mean and the variance.

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

The first row of the standard normal table, denoted as the z row, shows values of z up to the _____ decimal point.

hundredths, hundredth, or second

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

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

A random variable X follows the continuous uniform distribution if

it has an equally likely chance of assuming any value within a specified range.

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

one

The normal distribution is described by two parameters

the population mean μ and the population variance

For a discrete random variable X,

there are a countable number of possible values.

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

zero

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

zero

For a continuous random variable X, how many distinct values can it assume over an interval?

Uncountably Infinite

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−μ/σ

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.

Since the z table provides the cumulative probabilities for a given value of z, how can we calculate P(Z > z)?

= 1 - P(Z ≤ z)

Standard transformation

A normally distributed random variable X with mean μ and standard deviation σ can be transformed into the standard normal random variable Z as Z = (X - μ)/σ.

Standard normal distribution

A special case of the normal distribution with a mean equal to zero and a standard deviation (or variance) equal to one.

THE STANDARD TRANSFORMATION: CONVERTING X INTO Z

Any normally distributed random variable X with mean μ and standard deviation σ can be transformed into the standard normal random variable Z as

The normal probability distribution, or simply the normal distribution, is also referred to as the

Gaussian distribution

Which of the following is an example of a continuous random variable?

Normal random variable

The normal distribution is

asymptotic

For a continuous random variable X, the number of possible values

cannot be counted.

The inverse transformation, x = μ + zσ is used to ______.

compute x values for given probabilities

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

The z table used in the textbook provides______ probabilities.

cumulative

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

f(x) ≥ 0 for all values x of X.

A continuous random variable X follows the uniform distribution with a lower limit of a and an upper limit of b. The ______ of X is calculated using the formula √(b−a)2/12.

standard deviation

A normal random variable X is transformed into Z by ______.

subtracting the mean, and then dividing by the standard deviation.

Rate of return on an investment follows the --- distribution.

normal

Continuous uniform distribution

A distribution describing a continuous random variable that has an equally likely chance of assuming a value within a specified range.

THE CONTINUOUS UNIFORM DISTRIBUTION

A random variable X follows the continuous uniform distribution if its probability density function is where a and b represent the lower limit and the upper limit, respectively, that the random variable assumes.

Examples of random variables that closely follow a normal distribution include:

Salary of employees in a tech firm Scores on the SAT exam Cumulative debt of college graduates Advertising expenditure of firms Rate of return on an investment the weights of newborn babies.

THE PROBABILITY DENSITY FUNCTION

The probability density function f(x) for a continuous random variable X has the following properties: f(x) ≥ 0 for all possible values x of X, and the area under f(x) over all values x of X equals one.

THE STANDARD NORMAL DISTRIBUTION

The standard normal random variable Z is a normal random variable with E(Z) = 0 and SD(Z) = 1. The z table provides cumulative probabilities P(Z ≤ z) for positive and negative z values.

The normal distribution is

bell-shaped and symmetric around its mean