Section 7.2 Part 1 – Means and Variances of Random Variables
Means and Variances of Random Variables
Probability is the mathematical language that describes the long-run regular behavior of random phenomena.
The probability distribution of a random variable is an idealized relative frequency distribution.
o See example 7.5 on p.407
Probability distribution of X:
Payoff X: $0 $500
Probability: 0.999 0.001
Mean of the random variable X is found by:
In other words, you are expected to lose $0.50 per ticket if many tickets are purchased over time.
The mean, ____, of a ______is their ______.
The mean of a______is also a ______, but
with an essential change to take into account the fact that ______need to be ______.
Mean of a Discrete Random Variable
Suppose that X is a discrete random variable whose distribution is
Value of X: x1 , x2 ,x3 , …. xk
probability: p1 , p2 , p3 , ….. pk
To find the mean of X, multiply each possible value by its probability, then add all products; = x1 p1 + x2 p2+ …….xk pk =∑xi pi Mean and Expected Value
The mean of a probability distribution describes the long-run average outcome.
You will often find the mean of a random variable X called expected value of X.
The common symbol _____, the Greek letter mu, is used to represent the mean of a probability distribution (expected value).
Some other common notations include:
(this is the most common)
The Variance of Random Variable
The mean is a measure of the center of a distribution.
The variance and the standard deviation are the measures of ______the choice of
______to measure center.
Recall from chapter 2 that the variance of a data set is written as ______and it represents an average of the squared deviation from the mean.
To distinguish between the variance of a data set and the variance of a random variable X, we write the variance of a random variable X as
Definition:
Suppose that X is a discrete random variable whose probability distribution is
Value: x1 x2 x3 …
Probability: p1 p2 p3 …
and that µX is the mean of X. The variance of X is
The standard deviation ______of X is the ______Example 7.7 – Selling Aircraft Parts
Gain Communications sells aircraft communications units to both the military and the civilian markets. Next year’s sales depend on market conditions that cannot be predicted exactly. Gain follows the modern practice of using probability estimates of sales. The military division estimates its sales as follows:
Units sold: 1000 3000 5000 10,000
Probability: 0.1 0.3 0.4 0.2
Calculate the mean and variance of X
See p.411 to check your answers
Statistical Estimation and the Law of Large Numbers
To estimate μ, we choose a SRS of young women and ______the unknown
______.
Statistics obtained from ______are ______because their
values would ______.
It seems reasonable to use ______to estimate ______.
A SRS should fairly represent the ______, so the mean of the sample should be
______μ of the population.
Of course, we don’t expect to be exactly equal to μ, and realize that if we choose another SRS, the luck of the draw will probably produce a different .
Law of Large Numbers
If we keep on adding observations to our random sample, the statistic is guaranteed to get
______to the ______and then ______.
This remarkable fact is called the law of large numbers.
The law of large numbers states the following:
Draw independent observations at random from any population with finite mean μ.
Decide how ______you would like to estimate μ.
As the number of observations draw ______, the mean of the observed values eventually
______of the population ______and then stays that close. See example 7.8 on p.414 The “Law of Small Numbers”
Both the rules of probability and the law of large numbers describe the regular behavior of chance phenomena in the long run.
Psychologists have discovered most people believe in an ______“law of small numbers”
That is, we expect even short sequences of random events to show the kind of average behavior that in fact appears only in the long run.
How Large is a Large Number?
The law of large numbers says that the actual mean outcome of many trials gets close to the distribution mean μ as more trials are made.
It doesn’t say how many trials are needed to guarantee a mean outcome close to μ.