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If A and B are mutually exclusive events with P(A) = 0.70, then P(B)
can be any value between 0 and 1.
can be any value between 0 and 0.70.
cannot be larger than 0.30.
None of the above statements are true.
If A and B are independent events with P(A) = 0.20 and P(B) = 0.60, then P(A|B) is
0.2000
In the notation below, X is the random variable, c is a constant, and V refers to the variance. Which of the
following laws of variance is not correct?
V(c) = 0
V(X + c) = V(X)
V(X + c) = V(X) + c
V(cX) = c2 V(X)
Which of the following statements is always correct?
P(A and B) = P(A) * P(B)
P(A or B) = P(A) + P(B)
P(A or B) = P(A) + P(B) + P(A and B)
P ( AC ) = 1- P(A)
follows: P(
A1 and A2 , and two columns, B1 and B2 , are as
A1 and B1 ) = .10, P( A1 and B2 ) = .30, P( A2 and B1 ) = .05, and P( A2 and B2 ) = .55. Then P( A1 )
0.40.
If the random variable X follows a Poisson distribution with 
 7.5 , what is P(X>5)?
0.868
An effective and simple method of applying the basic probability rules is the
probability tree.
Suppose Z is a random variable with a standard normal distribution.
What proportion of observations from Z are greater than -0.13?
z-0.13 = 0.4483
P(z>-0.13) = 1-0.4483 = 0.5517
What is P(0.29<Z<1.45)?
z1.45 = 0.9265, z0.29 = 0.6141
P(0.29Z1.45) = 0.9265-0.6141 = 0.3164
An experiment consists of tossing an unbiased coin three times. Drawing a probability tree for this experiment
will show that the number of simple events in this experiment is
8
Use the following information to answer questions 6 and 7: The weights of newborn children in the United
States vary according to a normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The
government classifies a newborn as having low birth weight if the weight is less than 5.5 pounds.
What proportion of babies weigh less than 5.5 pounds at birth?
0.0548
If the government wanted to change the value 5.5 pounds to a weight where only 2% of newborns weigh less
than the new value, what weight should they use?
4.9375 pounds
If you are given a table of joint probabilities of two events, any probability computed by adding across rows or
down columns is also called
marginal probability
joint probability
conditional probability
Bayes’ theorem
The effect of increasing the standard deviation of a normally distributed random variable is that the distribution
becomes
narrower and more peaked.
flatter and wider.
If the random variable X follows a Uniform distribution with a=7.5 and b=11.5, what is P(X<10)?
0.625
The weighted average of the possible values that a discrete random variable X can assume, where the weights
are the probabilities of occurrence of those values, is referred to as the
variance.
standard deviation.
expected value.
covariance.
The variance of a binomial distribution for which n = 100 and p = 0.20 is
16.
A professor receives, on average, 28.6 e-mails from students the day before the final exam. To compute the
probability of receiving at least 8 e-mails on such a day, he will use what type of probability distribution?
Binomial distribution
Poisson distribution
Normal distribution
Exponential distribution
Given that X is a normally distributed random variable, which of the following statements is true?
The variable X + 5 is also normally distributed.
The variable X - 5 is also normally distributed.
The variable 5X is also normally distributed.
All of the above.
If two events are mutually exclusive, what is the probability that both occur at the same time?
0.00
The chancellor of a major university was concerned about alcohol abuse on her campus and wanted to find out
the percentage of students at her university who visited city bars every weekend. Her advisor took a random
sample of 250 students. The percentage of students in the sample who visited city bars every weekend is an
example of
a categorical random variable.
a discrete random variable.
a continuous random variable.
a parameter.
If X and Y are any random variables, which of the following identities is not always true?
E (X+Y) = E(X) + E(Y)
V(X+Y) = V(X) + V(Y)
E(4X+5Y) = 4E(X) + 5 E(Y)
V(4X+5Y) = 16V(X) + 25V(Y) + 40COV(X,Y)
If the random variable X follows an exponential distribution with
0.3935
The joint probabilities shown in a table with two rows,
  .05 , what is P(X<10)?
What is the twentieth percentile of Z? [In other words, find the value of z such that approximately 20% of
standard normal observations fall below it.]
P(Z -0.84)
What is the third quartile of Z? [In other words, find the value of z such that the proportion of standard normal
observations falling below z is approximately 75%.]
0.67
In a shipment of 15 room air conditioners, there are 4 with defective thermostats. Two air conditioners will be
selected at random (without replacement) and inspected one after the other. Find the probability that: (Hint: A
probability tree may help.)
The first is defective.
P(1st defective = 4/15 = 0.267)
The first is defective and the second one is good.
P(1st D and 2nd G) = 0.79 * 0.27 = 0.2133
Both are defective.
P( 1st D and 2nd D) = 0.27 * 0.21 = 0.0567
The second one is defective, given that the first one was good.
P( 2nd D | 1st G) = 4/14 = 0.2857
Exactly one is defective.
P(exactly one D) = P( 1st D or 2nd D) = P(1st D) + P(2nd D) = 0.2667 + 0.2667 = 0.5334
QUIZZES CH6- Which of the following statements is always correct?
A. P(A and B) = P(A)P(B)
B. P(A or B) = P(A) + P(B)
C. P(A or B) = P(A) + P(B) + P(A and B)
D. P (Ac) = 1-P(A)
If a coin is tossed three times and a statistician predicts that the probability of obtaining three heads in a row is
0.125, which of the following assumptions is irrelevant to his prediction?
A. The events are dependent.
B. The events are independent.
C. The coin is unbiased.
D. All of these choices.
The manager of the customer service division of a major consumer electronics company is interested in
determining whether the customers who have purchased a videocassette recorder made by the company over
the past 12 months are satisfied with their products. The possible responses to the question "How many
videocassette recorders made by other manufacturers have you used"? are values from a:
discrete random variable.
continuous random variable.
categorical random variable.
parameter.
If X and Y are any random variables with E(X) = 50, E(Y) = 6, E(XY) = 21, V(X) = 9 and V(Y) = 10, then the
relationship between X and Y is a:
strong positive relationship.
strong negative relationship.
weak positive relationship.
weak negative relationship.
The manager of the customer service division of a major consumer electronics company is interested in
determining whether the customers who have purchased a videocassette recorder made by the company over
the past 12 months are satisfied with their products. The possible responses to the question "How many
videocassette recorders made by other manufacturers have you used"? are values from a:
categorical random variable.
continuous random variable.
parameter.
discrete random variable
If X and Y are random variables with E(X) = 5 and E(Y) = 8, then E(2X + 3Y) is:
40
13
18
34
A function or rule that assigns a numerical value to each simple event of an experiment is called:
a sample space.
a probability tree.
a probability distribution.
a random variable
i.
The random variable of a binomial experiment is defined as the number of
successes in the n trials. It is called the binomial random variable.
ii.
Must REPLACE the card to be binomial- if you don’t they’re not independ.
e.
The probability of x successes in a binomial experiment with n trials and probability of
success = p is
P(x)=……………….
f.
Finding the binomial probability P(X>x) = 1-P(X<[x-1])
g.
Finding the probability P(X=x) = P(X<x)-P(X<[x-1])
h.
Mean=µ=np; Var=δ2=np(1-p); SD=δ= square root of np(1-p)
3)
Poisson random variable- the number of successes that occur in a period of time or an interval of space;
number of occurrences of a relatively rare event that occurs randomly and independently
a.
P(x) = ……………………
Where µ is the mean number of successes in the interval or region and e is the bas of the
natural logarithm (approx 2.71828)
b.
Finding the Poisson Probability P(X>x)=1-P(X<[x-1])
c.
Finding the Poisson Probability P(X=x)= P(X<x) - P(X<[x-1])
CH8- continuous random variables
1)
Because there is an infinite number of values, the probability of each individual is virtually 0. We can
only determine the probability of range values.
2)
Requirements for a Probability Density Function whose randge is a<x<b
a.
F(x)>0 for all x between a and b
b.
The total area under the curve between a and b is 1
3)
Uniform (rectangular) density function f(x) 1/b-a where a<x<b
a.
P(x1<X<x2)=base x height = (x2-x1) x 1/b-a
4)
Normal Probability Density Function (of a normal random variable)
5)
Calculating normal probabilities standard normal variable Z=X-µ/δ
6)
Exponential Probability Density Function- a random variable X is exponentially distributed if its
probability density function is given by f(x)=
a.
Probabilities associated with an Exponential Random Variable
P(X>x) =
P(X<x) =
P(x1<X<x2) =P(X<x2) – P(X<x1) =
If X is a binomial random variable with n = 25, and p = 0.25, then the variance and standard deviation of X are
6.25 and 2.5, respectively.
True
False
The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.
True
False
CH6 REC WS- Assume that the weights of bags of potato chips are independent of one another. Assume that
the third quartile is 19 oz. Suppose I randomly select 4 bags of chips. P(W>18)=1/4, P(W<18)=3/4
What is the probability that all four bags weigh more than 18 oz?
P(All 4 bags > 18) = P(1st > 18) P(2st > 18) P(3st > 18) P(4st > 18)
What is the probability that none of the 4 bags weigh more than 18 oz?
P(NONE>18) + P(ALL < 18) = P(1st < 18) P(2st < 18) P(3st < 18) P(4st < 18)
What is the probability that at least one of the 4 bags weighs more than 18 oz?
P(At least 1 bag > 18) = 1 – P(NONE>18) = 1 – (¾)4
CH6
1)
2)
3)
4)
5)
6)
1)
2)
3)
4)
1)
2)
3)
CH7
1)
2)
3)
1)
2)
A random experiment is an action or process that leads to one of several possible outcomes
A sample space of a random experiment is a list of all possible outcome of the experiment. The
outcomes must me exhaustive (all possible outcomes must be included) and mutually exclusive (no two
outcomes can occur at the same time).
Requirements of probabilities: Given a sample space S={O1,O2,…,Ok}, the probabilities assigned to the
outcomes must satisfy two requirements:
a.
The probability of any outcome must lie between 0 and one 0<P(Oi)<1 for each i
b.
The sum of the probabilities of all the outcomes in a sample space must be one.
Classical approach is associated with games of chance; relative frequency approach defines probabilities
as the long-run frequency with which an outcome occurs; subject approach defines probability as the
degree of belief that we hold in the occurrence of an event
An individual outcome of a sample space is called a simple event; an event is a collection or set of one or
more simple events in a sample space
The probability of an event is the sum of probabilities of the simple events that constitute the event
Intersection of events A and B- the intersection is the event that occurs when both A and B occur
(denoted as: A and B); the prob. Of the int. is called the joint probability
Conditional probability- the probability of event A given event B is P(A|B)=P(A and B)/P(B)
Independent events- two events A and B are said to be independent if P(A|B)=P(A)
The union of events A and B is the event that occurs when either A or B or both occur.
Complement Rule- P(AC)=1-P(A)
The multiplication rule is used to calculate the joint probability of two events. P(A and B)= P(A|B)P(B)
a.
The joint probability of any two independent events A and B is P(A and B)=P(A)P(B)
The addition rule calculate the probability of the union of two events P(A or B)=P(A)+P(B)-P(A and B)
a.
The probability of the union of two mutually exclusive events A and B is
P(A or B)=P(A)+P(B)
A random variable is a function or rule that assigns a number to each outcome of an experiment often
labeled as (X). The number of heads when a coin is tossed is the random variable or value of X.
a)
Discrete random variable- one that can take on a countable number of values
i)
0<P(x)<1
for all x
ii)
 (of all x) P(x)=1
b)
Continuous random variable0 values are uncountable
i)
Probability distribution- table, formula, or graph that describes the values of a RV and the
probability associated with these values
Statistical inference deals with inferences about populations
a)
The population mean (expected value) is the weighted average of all its values, the weights are
the probabilities E(X)=µ=  xP(x)
i)
=0P(0)+1P(1)+2P(2)+…. nP(n)
b)
Population variance is the weighted average of the squared deviations from the mean
2
V(X)= δ =  (x-µ) 2P(x) or V(X)= δ2=  x 2P(x)-µ2
i)
=(0-µ) 2 P(0)+ (1-µ) 2 P(1)+ … (n-µ) 2 P(n) or = 02 P(0)+ 12 P(1) … n2 P(n) - µ2
c)
Standard Deviation is δ=square root of δ2
Laws of Expected Value
Laws of Variance
a)
E(c)=c
V(c)=0
b)
E(X+c)=E(X)+c
V(X+c)=V(X)
c)
E(cX)=cE(X)
V(cX)=c2V(X)
Bivariate distribution- probabilities combinations of two variables. The joint probability that the two
variables will assume the values x and y is denoted P(x, y).
a.
A bivariate or joint probability distribution of X and Y is a table or formula that lists the
joint probabilities for all pairs of values of X and Y.
b.
Requirements for discrete bivariate distribution: 0<P(x, y) < 1 for all pairs of values (x,y)
and  (all x) (all y) P(x,y)=1
i.
Covariance
ii.
Coefficient of Correlation
c.
Sum of 2 variables P(X+Y=2)=P(0,2)+P(1,1)+P(2,0)=.07+.06+.06=.19
x+y | 0 1 2 3 4
P(x+y)| .12 .63 .19 .05 .01
d.
Laws of expected value
Variance of the sum of two variables
E(X+Y)=E(X)+E(Y)
V(X+Y)=V(X)=V(Y)+2COV(X,Y)
*If x and y are ind., COV (X,Y)=0 and thus V(X+)=V(X)+V(Y)
Binomial experiment - discrete
a.
Consists of a fixed number of trials. We represent the number of trials by n.
b.
On each trial there are two possible outcomes. We label one outcome as a success, and
the other as a failure
c.
The probability of success is p. The probability of failure is 1-p.
d.
The trials are independent, which means that the outcome of one trial does not affect the
outcome of any other trial.
2