1. Discrete Random Variables and Probability Distributions3
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2. Population versus sample
Population: The entire group of individuals in which we are interested but can’t usually assess directly.
Example: All humans, all working-age people in California, all crickets
A parameter is a number describing a characteristic of the population.
Sample: The part of the population we actually examine and for which we do have data.
How well the sample represents the population depends on the sample design.
A statistic is a number describing a characteristic of a sample.
Population
Sample
Before we learn the experimental design, we need to distinguish between population and sample. As long as it is impossible to collect the data from all population, we do a sampling. We need to select the proper sample to represent the population well. A parameter is a fixed number, but in practice we do not know its values. We can estimate the parameter by a statistics. The value of statistic is known when we have taken a sample. But it can change from sample to sample. For example, we are interested in the proportion of the population who smokes in this country. Then the parameter is a fixed number. Suppose we check the smoking behavior of all american. Then we can find the proportion. However, it is impossible. That’s why we need a sample. If we investigate the behavior of 1000 people, we will find the proportion of this sample. This value is a statistic and we can estimate the parameter by this value. This statistic depends on sample.

3. Parameters and Statistics
As we begin to use sample data to draw conclusions about a wider population, we must be clear about whether a number describes a sample or a population.
A parameter is a number that describes some characteristic of the population. In statistical practice, the value of a parameter is not known because we cannot examine the entire population.
A statistic is a number that describes some characteristic of a sample. The value of a statistic can be computed directly from the sample data. We often use a statistic to estimate an unknown parameter.
We write µ (the Greek letter mu) for the population mean and σ for the population standard deviation. We write (x-bar) for the sample mean and s for the sample standard deviation.

4. Discrete random variables
A random variable is a variable whose value is a numerical outcome of a random phenomenon.
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops with the open side up.
A discrete random variable X has a finite or countable infinite number of possible values.
A bottle cap is tossed three times. The number of times the cap drops with the open side up is a discrete random variable (X). X can only take the values 0, 1, 2, or 3.

5. p (x) = P(X = x) = P (all s  S : X (s) = x),
Probability distribution for discrete random variables
The probability distribution of a random variable X lists the values and their probabilities:
The probabilities pi must add up to 1.
Definition of Probability distribution or probability mass function (pmf) of a discrete rv:
p (x) = P(X = x) = P (all s  S : X (s) = x),
for every number x.

6. Example:
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops with the open side up. The open side is lighter than the closed side so it will probably be up well over 50% of the tosses. Suppose on each toss the probability it drops with the open side up is 0.7.
Example: P(UUU) = P(U)* P(U)* P(U) = (.7)*(.7)*(.7) = .343 U
U UUU
D … D
D UUD
U UDU
D DDU …
Value of X
Probability
UDD UUD
DUD UDU
DDD DDU UUD UUU

7. The probability of any event is the sum of the probabilities pi of the values of X that make up the event.
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops with the open side up.
What is the probability that at least two times the cap lands with the open side up (“at least two” means “two or more”)?
Value of X
Probability
UDD UUD
DUD UDU
DDD DDU UUD UUU
P(X ≥ 2) = P(X=2) + P(X=3) = = 0.784
What is the probability that cap lands with the open side up fewer than three times?
P(X<3) = P(X=0) + P(X=1) + P(X=2) = = or
P(X<3) = 1 – P(X=3) = = 0.657

8. A Parameter of a Probability Distribution
The pmf of the Bernoulli rv X was p(0) = .8 and p(1) = .2 because 20% of all purchasers selected a desktop computer. At another store, it may be the case that p(0) = .9 and p(1) = .1.
More generally, the pmf of any Bernoulli rv can be expressed in the form p (1) =  and p (0) = 1 – , where 0 <  < 1. Because the pmf depends on the particular value of  we often write p (x; ) rather than just p (x):
(3.1)

9. Example:
Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born.
Let p = P (B), assume that successive births are independent, and define the rv X by x = number of births observed.
Then
p(1) = P(X = 1)
= P(B)
= p

10. Example: p(2) = P(X = 2) = P(GB) = P(G)  P(B) = (1 – p)p and
= P(GGB)
= P(G)  P(G)  P(B)
= (1 – p)2p

11. Example: Continuing in this way, a general formula emerges:
The parameter p can assume any value between 0 and 1.

12. The Cumulative Distribution Function
For some fixed value x, we often wish to compute the probability that the observed value of X will be at most x.
The probability that X is at most 1 is then
P(X  1) = p(0) + p(1) = = .667

13. The Cumulative Distribution Function
In this example, X  1.5 if and only if X  1, so
P(X  1.5) = P(X  1) = .667
Similarly,
P(X  0) = P(X = 0) = .5, P(X  .75) = .5
And in fact for any x satisfying 0  x < 1, P(X  x) = .5.

14. The Cumulative Distribution Function
The largest possible X value is 2, so
P(X  2) = 1 P(X  3.7) = 1 P(X  20.5) = 1
and so on.
Notice that P(X < 1) < P(X  1) since the latter includes the probability of the X value 1, whereas the former does not.
More generally, when X is discrete and x is a possible value of the variable, P(X < x) < P(X  x).

15. The Cumulative Distribution Function
Definition of cumulative distribution function (cdf) F(x) of a discrete rv variable X with pmf p(x):
F (x) = P(X  x) = ,
for every number x.
For any number x, F(x) is the probability that the observed value of X will be at most x.

16. Example:
A store carries flash drives with either 1 GB, 2 GB, 4 GB, 8 GB, or 16 GB of memory.
The accompanying table gives the distribution of Y = the amount of memory in a purchased drive:

17. Example:
Let’s first determine F (y) for each of the five possible values of Y:
F (1) = P (Y  1)
= P (Y = 1)
= p (1)
= .05
F (2) = P (Y  2)
= P (Y = 1 or 2)
= p (1) + p (2)
= .15

18. Example: F(4) = P(Y  4) = P(Y = 1 or 2 or 4) = p(1) + p(2) + p(4)
cont’d
F(4) = P(Y  4)
= P(Y = 1 or 2 or 4)
= p(1) + p(2) + p(4)
= .50
F(8) = P(Y  8)
= p(1) + p(2) + p(4) + p(8)
= .90
F(16) = P(Y  16)
= 1

19. Example:
Now for any other number y, F (y) will equal the value of F at the closest possible value of Y to the left of y. For example,
F(2.7) = P(Y  2.7)
= P(Y  2)
= F(2)
= .15
F(7.999) = P(Y  7.999)
= P(Y  4)
= F(4)
= .50

20. Example:
If y is less than 1, F (y) = 0 [e.g. F(.58) = 0], and if y is at least 16, F (y) = 1[e.g. F(25) = 1]. The cdf is thus

21. Example: A graph of this cdf is shown in Figure 3.5.
A graph of the cdf of Example 3.13
Figure 3.13

22. The Cumulative Distribution Function
For X a discrete rv, the graph of F (x) will have a jump at every possible value of X and will be flat between possible values. Such a graph is called a step function.
Proposition
For any two numbers a and b with a  b,
P (a  X  b) = F (b) – F (a–)
where “a–” represents the largest possible X value that is strictly less than a.

23. Expected value of a random variable
The expected value of a random variable X is also called mean of X.
The mean x bar of a set of observations is their arithmetic average.
The mean µ of a random variable X is a weighted average of the possible values of X, reflecting the fact that all outcomes might not be equally likely.
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops with the open side up (“U”).
UDD UUD
DUD UDU
DDD DDU UUD UUU
Value of X
Probability

24. Expected value of a random variable
Definition:
Let X be a discrete random variable with set of possible values D and pmf p (x). The expected value or mean value of X, denoted by E(X) or X or just , is

25. Mean of a discrete random variable
For a discrete random variable X with probability distribution 
the mean µ of X is found by multiplying each possible value of X by its probability, and then adding the products.
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops with the open side up.
The mean µ of X is
µ = (0*.027)+(1*.189)+(2*.441)+(3*.343)
= 2.1
Value of X
Probability

26. Variance of a random variable
The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center.
The variance σ2X of a random variable is a weighted average of the squared deviations (X − µX)2 of the variable X from its mean µX. Each outcome is weighted by its probability in order to take into account outcomes that are not equally likely.
The larger the variance of X, the more scattered the values of X on average. The positive square root of the variance gives the standard deviation σ of X.

27. Variance of a random variable
Definition Let X have pmf p (x) and expected value . Then the variance of X, denoted by V(X) or 2X , or just 2, is
The standard deviation (SD) of X is

28. Variance of a discrete random variable
For a discrete random variable X with probability distribution 
and mean µX, the variance σ2 of X is found by multiplying each squared deviation of X by its probability and then adding all the products.
A bottle cap is tossed three times. We define the random variable X as the number of number of times the cap drops
with the open side up. µX = 2.1.
Value of X
Probability
The variance σ2 of X is
σ2 = .027*(0−2.1) *(1−2.1) *(2−2.1) *(3−2.1)2
= = .63

29. A Shortcut Formula for 2
The number of arithmetic operations necessary to compute 2 can be reduced by using an alternative formula. Proposition:
V(X) = 2 = – 2 = E(X2) – [E(X)]2
In using this formula, E(X2) is computed first without any subtraction; then E(X) is computed, squared, and subtracted (once) from E(X2).

30. Rules for means and variances
If X is a random variable and a and b are fixed numbers, then
µa+bX = a + bµX
σ2a+bX = b2σ2X
If X and Y are two independent random variables, then
µX+Y = µX + µY
σ2X+Y = σ2X + σ2Y

31. $$$
Investment
You invest 20% of your funds in Treasury bills and 80% in an “index fund” that represents all U.S. common stocks. Your rate of return over time is proportional to that of the T-bills (X) and of the index fund (Y), such that R = 0.2X + 0.8Y.
Based on annual returns between 1950 and 2003:
Annual return on T-bills µX = 5.0% σX = 2.9%
Annual return on stocks µY = 13.2% σY = 17.6%
Assume that X and Y are independent
µR = 0.2µX + 0.8µY = (0.2*5) + (0.8*13.2) = 11.56%
σ2R = σ20.2X + σ20.8Y = 0.2*2σ2X + 0.8*2σ2Y
= (0.2)2(2.9)2 + (0.8)2(17.6)2 =
σR = √ = 14.09%
The portfolio has a smaller mean return than an all-stock portfolio, but it is also less risky.

32. Binomial distributions
Binomial distributions are models for some categorical variables, typically representing the number of successes in a series of n trials.
The observations must meet these requirements:
The total number of observations n is fixed in advance.
Each observation falls into just 1 of 2 categories: success and failure.
The outcomes of all n observations are statistically independent.
All n observations have the same probability of “success,” p.
Population values were quantitative. We will now shift our focus to population values that are categorical. Particularly, we focus on the simplest case of a random variable with only two possible categories. What is binomial distribution? The binomial distribution is a new family of probability distributions.
We record the next 50 births at a local hospital. Each newborn is either a boy or a girl; each baby is either born on a Sunday or not.

33. The parameter n is the total number of observations.
We express a binomial distribution for the count X of successes among n observations as a function of the parameters n and p: B(n,p).
The parameter n is the total number of observations.
The parameter p is the probability of success on each observation.
The count of successes X can be any whole number between 0 and n.
A coin is flipped 10 times. Each outcome is either a head or a tail. The variable X is the number of heads among those 10 flips, our count of “successes.”
On each flip, the probability of success, “head,” is 0.5. The number X of heads among 10 flips has the binomial distribution B(n = 10, p = 0.5).

34. Applications for binomial distributions
Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations.
They are used when we want to know about the occurrence of an event, not its magnitude.
In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel.
Is a person ambitious or not? The binomial distribution describes the number of ambitious persons, not how ambitious they are.
In quality control we assess the number of defective items in a lot of goods, irrespective of the type of defect.

35. Binomial distribution in statistical sampling
A population contains a proportion p of successes. If the population is much larger than the sample, the count X of successes among size n has approximately the binomial distribution B(n, p).
The n observations will be nearly independent when the size of the population is much larger than the size of the sample. As a rule of thumb, the binomial distribution can be used when the population is at least 20 times as large as the sample.

36. Binomial mean and standard deviation
The center and spread of the binomial distribution for a count X are defined by the mean m and standard deviation s: a) b)
We often write q as 1 – p.
Effect of changing p when n is fixed.
a) n = 10, p = 0.25
b) n = 10, p = 0.5
c) n = 10, p = 0.75
For small samples, binomial distributions are skewed when p is different from 0.5.
Sampling distribution for counts. c)

37. Color blindness
The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is estimated to be about 8%. We take a random sample of size 25 from this population.
The population is definitely larger than 20 times the sample size, thus we can approximate the sampling distribution by B(n = 25, p = 0.08).
What are the mean and standard deviation of the count of color blind individuals in the SRS of 25 Caucasian American males?
µ = np = 25*0.08 = 2
σ = √np(1  p) = √(25*0.08*0.92) = 1.36

38. Calculations for binomial probabilities
The binomial coefficient counts the number of ways in which k successes can be arranged among n observations.
The binomial probability P(X = k) is this count multiplied by the probability of any specific arrangement of the k successes: X
P(X) 1 2 … k n
nC0 p0qn = qn
nC1 p1qn-1
nC2 p2qn-2
nCx pkqn-k
nCn pnq0 = pn
Total
The probability that a binomial random variable takes any range of values is the sum of each probability for getting exactly that many successes in n observations.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

39. Binomial formulas
The number of ways of arranging k successes in a series of n observations (with constant probability p of success) is the number of possible combinations (unordered sequences).
This can be calculated with the binomial coefficient:
Where k = 0, 1, 2, ..., or n.

40. n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1
Binomial formulas
The binomial coefficient “n_choose_k” uses the factorial notation “!”.
The factorial n! for any strictly positive whole number n is:
n! = n × (n − 1) × (n − 2) × · · · × 3 × 2 × 1
For example: 5! = 5 × 4 × 3 × 2 × 1 = 120
Note that 0! = 1.

41. Color blindness
The frequency of color blindness (dyschromatopsia) in the Caucasian American male population is estimated to be about 8%. We take a random sample of size 25 from this population.
What is the probability that exactly five individuals in the sample are color blind?
P(x = 5) = 53,130 * * =