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100 Chapter 2 Review of Probability

2. The expected value of a random variable Y (also called its mean, mY),
denoted E(Y), is its probability-weighted average value. The variance of Y
is sY2 = E3(Y - mY)24, and the standard deviation of Y is the square root
of its variance.

3. The joint probabilities for two random variables X and Y are summarized
by their joint probability distribution. The conditional probability distribu-
tion of Y given X = x is the probability distribution of Y, conditional on X
taking on the value x.

4. A normally distributed random variable has the bell-shaped probability
density in Figure 2.5. To calculate a probability associated with a normal
random variable, first standardize the variable and then use the standard
normal cumulative distribution tabulated in Appendix Table 1.

5. Simple random sampling produces n random observations Y1, c, Yn that
are independently and identically distributed (i.i.d.).

6. The sample average, Y, varies from one randomly chosen sample to the next
and thus is a random variable with a sampling distribution. If Y1, c, Yn are
i.i.d., then:

a. the sampling distribution of Y has mean mY and variance sY2 = s2Y>n;
b. the law of large numbers says that Y converges in probability to mY; and

c. the central limit theorem says that the standardized version of Y,
(Y - mY)>sY, has a standard normal distribution 3N(0, 1) distribution]
when n is large.

Key Terms

outcomes (61) probability density
probability (61) function (p.d.f.) (65)
sample space (61)
event (61) density function (65)
discrete random variable (61) density (65)
continuous random variable (61) expected value (65)
probability distribution (62) expectation (65)
cumulative probability mean (65)
variance (67)
distribution (62) standard deviation (67)
cumulative distribution function moments of a distribution (69)
skewness (69)
(c.d.f.) (63) kurtosis (71)
Bernoulli random variable (63) outlier (71)
Bernoulli distribution (63)