1.
Set
a.
b.
2.
3.
4.
Definitions
a. Random Experiment: An experiment that can result in different outcomes, even though it is performed under the
same conditions and in the same manner.
b. Sample Space: This is the set of all possible outcomes.
i. Discrete: Consists of finite or countable infinite set of outcomes
ii. Continuous: Contains an interval (either finite or infinite) of real numbers
c. Event: It is a subset of the sample space of a random experiment.
i. Mutually Exclusive:
Probability
( )
a.
( )
(
)
( )
( )
b.
(
)
(
)
( )
( )
( )
(
)
(
)
(
)
c.
(
)
( )
(
)
d.
(
)
)
Conditional Probability: (
(
a.
5.
6.
)
(
) ( )
(
( )
B
0.99
(
( )
( )
( )
) ( )
(
) ( )
Fig 2:- Shows the probability of Success
7.
a.
b.
c.
(
(
(
)
(
) ( )
(
(
) (
)
d.
(
)
( )
(
)
(
) ( )
)
(
( ) ( )
( )
C
0.8
( )
(
( ))(
) ( )
( ))
(
)(
)
( ) ( )
(
) ( )
) ( )
) ( )
(
) (
)
Mean (Expected Value):
( )
9.
)
( )
( )
(
) ( )
(
) ( )
b.
Independence: Two events (A & B) are independent if one of the conditions listed below is satisfied;
(
)
a.
( )
(
)
( )
b.
(
)
( ) ( )
c.
CIRCUIT PROBLEM:-
A
0.99
8.
(
)
∑
( )
Variance:
( )
(
)
∑(
)
( )
∑
( )
10. Standard Deviation:
√
11. Discrete Uniform Distribution (DUD): This is when every member of the sample space has the same probability.
( )
a.
For DUD, the
( )
b.
For DUD, the
(
)
(
)
12. Bernoulli Trials:- A random experiment consists of Bernoulli trials if;
The trials are independent.
Each trial results in only two possible outcomes (Success and Failure).
The probability of success in each trial remains constant
a.
( )
( )
(
(
( )
( )
(
( )
Binomial Random Variable:)
)
)
(
A particularly long traffic light on your morning commute is
green 20% of the time that you approach it. Assume that each
morning represents an independent trial.
a) Over five mornings, what is the probability that the
light is green on exactly one day?
)
Page 1 of 6
b.
( )
(
(
( )
The probability of a successful optical alignment in the
assembly of an optical data storage product is 0.8. Assume the
trials are independent.
a) What is the probability that the first successful
alignment requires exactly four trials?
)
( )
(
)
( )
c.
( )
(
(
)
Negative Binomial:)
(
( )
)
( )
(
(
Geometric Random Variable:)
)
( )
)
13. Hypergeometric: ( )
(
(
(
)
)
A state runs a lottery in which six numbers are randomly
selected from 40, without replacement. A player chooses six
numbers before the state’s sample is selected.
a) What is the probability that the six numbers chosen
by a player match all six numbers in the state’s
sample?
)( )
(
( )
( )
( )
( )
( )
)(
)
(
(
)
)
14. Poisson Distribution:- This refers to any random experiment with the properties below;
Counting the number of occurrence of an event over a period of time/space.
The # of occurrence of an event in an interval is proportional to the length of the interval.
Events cannot occur simultaneously.
Occurrences of the events are independent for non-overlapping intervals.
( )
(
( )
)
( )
(
( )
)
( )
15. Continuous Random Variable (C.R.V)
( )
(
( )
)
( )
∫
(
)
( )
∫
( )
∫
16. Continuous Uniform Distribution:
( )
a.
For CUD, the
( ) (
b.
For CUD, the
(
)
)
(
(
)
)
17. Normal Distribution:
(
( )
a.
)
√
Note: you can only read from the table when
(
)
standardized…
( )
( )
( )
( )
(
18. Normal Approximation to Binomial Distribution: Use iff both
a.
(
)
)
√
b.
c.
(
(
)
(
(
))
(
)
√
(
)
Page 2 of 6
19. Normal Approximation to Poison RV: Use iff both
a.
√
20. Exponential Distribution: Amount of wait time until first count is obtained use integral.
( )
( )
(
)
∫
( )
(
)
∫
a.
(
(
(
( )
(
(
a.
(
( )
b.
)
(
)
( )
( )
(
)
(
( )
( )
( )
)
( ) ( )
√
( )
(
∑
)
( )
(
a) What is the mean time until a packet is
formed, that is, until five messages have
arrived at the node?
b) What is the probability that a packet is
formed in less than 10 seconds?
)
)
4-116. In a data communication system, several
messages that arrive at a node are bundled into
a packet before they are transmitted over the
network. Assume the messages arrive at the
node according to a Poisson process with
messages per minute. Five messages
are used to form a packet.
( )
(
)
What is the probability that EQ is detected before 2 years
if it doesn’t occur the first year? EQ occurs at a rate of 1.5
yearly.
)
21.
( )
(
∑
Lack of Memory Property:
)
)
)
)
(
)
22. Weibull Dist. :
(
( )
[( ) ( )
( )
(
(
4-135. Suppose the lifetime of a component (in
hours) is modeled with a Weibull distribution with
and
. Determine the following:
(a) (
)
(b) (
)
(c) value for x such that (
)
)
)
[
)
(
)
)
(
)
( )
( )
(
)
]
( )
)
)
(
( )
][
(
(
(
)
((
)
(
))
((
) (
))
(
(
)
(
)
(
(
)
(
)
)
( )
(
)
(
)
(
)
( )
)
))) ( )
( (
( )) (
)
)(
)
(
(
)
(
)
)
]
(
)
23. Lognormal Dist. :
)
( )
[
( )
(
][
√
(
)
( )
)
)
)
(
)
(
]
(
( )
[
(
(
( ( )
)
]
(
)
)
4-138. Suppose that X has a lognormal
distribution with parameters
and
. Determine the following:
)
(a) (
(
)
(b)
(c)
)
[
]
)
[
[
]
( )
(
(
[
)
]
]
( )
)
(
)
)
( ( )
)
(
)
24. Beta Dist. :
(
( )
(
)
(
( ) ( )
)
(
(
)
)
)
)
(
(
)
∫(
)
∫(
(
)
(
( ) ( )
(
)
(
( ) ( )
)
)
)
)
( )
4-150. Suppose X has a beta distribution
with parameters
and
.
Determine the following:
(a) (
)
)
(b) (
(c) mean and variance
)
(
) (
)
(
(
(
)
(
)
(
)
(
)
(
(
)
)
)
(
∫
(
)
(
[
(
(
( ) (
)
)
(
)
)
∫(
)
(
)
)
(
)
(
[
)
(
) )
∫(
(
(
)
)
(
(
)
( ) (
(
) )
)
∫(
)
)
]
)
)
(
)
]
Page 3 of 6
25. Joint Probability:
(
)
∫
(
∫
)
∑
(
)
(
( )
( )
(
)
( )
)
∑[(
( ))
∑ [(
(
∬
)
∑(
( )
]
)
∑(
( )
]
)
( )
∑[(
( ))
∑ [(
)
(
]
)
∑ [( )( )
(
]
)
[ (
)
]
( ) ( )]
)
(
( )
∫ ∫
(
)
( )
∫ ∫
(
)
( )
(
∫ ∫
( )
)
)
(
∫ ∫
)
(
)
(
)
(
∫ ∫
[ (
)
)
( ) ( )]
1. Marginal Probability:
( )(
)
( )
(
∑
(
∫
(
)
( )(
)
( )
(
)
(
∫ ∫
(
∑
(
∫
)
)
)
)
( )(
)
(
∫ ∫
( )(
)
)
)
〈
〈
(
)
(
(
)〉
)〉
2. Conditional Probability:
(
)
( )
( )
(
(
)
∫
)
( )
∫
(
( )
∫
( )
(
)
)
(
(
)
[∫
( )
)
]
(
(
[ (
)]
)
(
)(
) (
)
( )
[∫
) (
)
]
[ (
Covariance (
(
)
)]
):
(
)
Correlation (
) (
)
)
(
(
(
)
((
(
) (
)
(
) (
)
))
):
( ) ( )
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