Lecture 6

Continuous random variables

Continuous random variables

_______, such as heights and weights, length of life of a particular product, or experimental laboratory error, can assume the infinitely many values corresponding to points on a line interval or real line.

probability distribution of the continuous random variable

As the number of measurements becomes very large and the class widths become very narrow, the relative frequency histogram appears more and more like the smooth curve in (d). This smooth curve describes the _______.

area

The _______ under a continuous probability distribution is equal to 1.

area under the curve

The probability that X will fall into a particular interval - say, from a to b - is equal to the _______ between the two points a and b.

0

P(X = a) = _______ for continuous random variables

continuous uniform random variable

The _______ is used to model the behavior of a random variable whose values are uniformly or evenly distributed over a given interval.

1

1/b-a x (b-a) = _______

curve

The probability that X falls in an interval is calculated as the area under the _______ over the that interval.

b-a/2, 1/12(b-a)^2

The mean and variance of x are given by μ = _______ and σ^2 = _______

exponential random variable

The _______ is used to model positive continuous random variables such as waiting times or lifetimes associated with electronic components.

intensity

The parameter λ ( the Greek letter "lambda") is often referred to as the _______ and is related to the mean and variance as μ = 1/λ and σ^2 = 1/λ^2 so that μ = σ.

mold-shaped, normal

A large number of random variables observed in nature possess a frequency distribution that is approximately _______ and can be modeled by a _______ probability distribution.

center, symmetric

The mean μ locates the _______ of the distribution, and the distribution is _______ about its mean μ.

0.5, 0.5

Since the total area under the normal probability distribution is equal to 1, this symmetry implies that the area to the right of μ is _______ and the area to the left of μ is also _______.

sigma

The shape of the distribution is determined by _______, the population standard deviation.

height, spread, height, spread

Notice the differences in shape and location. Large values of σ reduce the _______ of the curve and increase the _______; small values of σ increase the _______ of the curve and reduce the _______.

standardize

To _______ a normal random variable X, we express its value as the number of standard deviations (σ) it lies to the left or right of its mean μ.

standard normal distribution

The probability distribution for z, shown in figure, is called the _______ because its mean is 0 and its standard deviation is 1.