Chapter 6

Continuous Uniform Distribution

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

standard normal distribution

E(Z)=0 and SD(Z)=1

Closely approximates the probability distribution of a wide range of random variables, such as the

Heights and weights of newborn babies Scores on SAT Cumulative debt of college graduates Serves as the cornerstone of statistical inference.

A continuous random variable has the uniform distribution on the interval [a, b] if its probability density function f(x) __________. Provides all probabilities for all x between a and b Is bell-shaped between a and b Is constant for all x between a and b, and 0 otherwise Asymptotically approaches the x axis when x increases to +∞ or decreases to -∞

Is constant for all x between a and b, and 0 otherwise

Continuous Random Variable

P(a<X<b) = P(a<=X<=b)

The Normal Distribution

Symmetric Bell-shaped asymptotic - tails get closer and closer to the horizontal axis but never touch it.

Continuous

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

Continuous Random Variable

When computing probabilities: P(X = x) = 0.

A continuous random variable

can be described by the probability density function and the cumulative distribution function.

A discrete random variable

can be described by the probability mass function and the cumulative distribution function.

Cumulative distribution functions

can only be used to compute probabilities for continuous random variables.

Continuous

can take on any value within an interval or collection of intervals

Which of the following is not a characteristic of a probability density function f(x)? f(x) ≥ 0 for all values of x. f(x) is symmetric around the mean. The area under f(x) over all values of x equals one. f(x) becomes zero or approaches zero if x increases to +infinity or decreases to -infinity.

f(x) is symmetric around the mean.

A continuous random variable

is characterized by uncountable values and can take on any value within an interval

The mean of a continuous uniform distribution

is simply the average of the upper and lower limits of the interval on which the distribution is defined.

The probability density function

of a continuous random variable is a counterpart to the mass function of a discrete random variable.

Cumulative Distribution Function

F(x) = P(X<=x)

probability density function

F(x)={1/(b-a) for a<=X<=b, and F(x)={0 for a > X > b, and

The cumulative distribution function F(x) of a continuous random variable X with the probability density function f(x) is which of the following? The area under f over all values x The area under f over all values that are x or less The area under f over all values that are x or more The area under f over all non-negative values that are x or less

The area under f over all values that are x or less

Discrete

The random variable assumes a countable number of distinct values.

Continuous Random Variable

Thus, since P(X = a) and P(X = b) both equal zero, the following holds for continuous random variables

Continuous Random Variable

We cannot assign a nonzero probability to each infinitely uncountable value and still have the probabilities sum to one.

The probability density function

of a continuous random variable is the counterpart to the probability mass function of a discrete random variable.

Standard Normal Table

referred to as z table

Standard Normal Variable

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

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 (b - a)/2 between a and b, and zero otherwise (a + b)/2 between a and b, and zero otherwise 1/(a + b) between a and b, and zero otherwise

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

probability density function

Describes the relative likelihood that X assumes a value within a given interval (e.g., P(a < X < b) ), where f(x) > 0 for all possible values of X. The area under f(x) over all values of x equals one.

probability density function

counterpart to probability mass function