Download HL Probability and Distributions PP

Warm Up 11.16.14
• The most common question asked was ‘why
does the variance have to be equal to the
mean?’ so let’s prove it!
• First you will need to explain why the
following is true:
• Use the above to demonstrate that E(X2) = λ2
+λ
• Now that you have the parts finish your proof
that Var(X) = λ
Warm Up #2
• The random variable X follows a Poisson
distribution with mean m and satisfies
P(X = 1) + P(X = 3) = P(X = 0) + P(X = 2)
(a)Find the value of m correct to four decimal
places.
(b)For this value of m, calculate P(1 ≤ X ≤ 2).
Test Next Class!
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5.2 SWBAT… Concepts of trial, outcome, equally likely outcomes,
sample space (U) and event; the probability of event A as
P(A)=n(A)n(U) ; complementary events A and A’; Venn diagrams, tree
diagrams, counting principles and tables of outcomes to solve
problems.
5.3 SWBAT… determine probabilities for combined events,
understand and utilize the formula for P(A⋃B), determine mutually
exclusive events.
5.4 SWBAT… calculated conditional probabilities, probabilities for
independent events and use Bayes’ theorem for a maximum of three
events.
5.5 SWBAT… Understand concepts of discrete and continuous
random variables and their probability distribution. Define and use
probability density functions. Find expected values (mean), mode,
median, variance and standard deviation. Apply random variables and
expected value.
5.6 SWBAT… Analyze situations by applying the binomial
distribution and examining the mean and its variance, apply Poisson
distribution and utilize its mean and variance
5.7 SWBAT… Understand the Normal distribution, its properities and
the standardized normal values
Homework – IA’s!!!
• Come prepared with IA topic and why you
are interested in the topic
• If still trying to figure out what you want to
do then come with a list of things you are
personally interested in
Normal Answers
•
1)
2)
3)
4)
•
1)
P.2
68-95-99.7
a. 16% b. 84% c. 99.7% d.0.15%
X~N(184,5)
PDF finds exact value, CDF finds range
P.3
a. Physics: -0.463
Chem: 0.431 Maths: 0.198
German: 0.521
Bio: -0.769
b. G, C, M, P, B
2) Inverse does the reverse process, takeing you from the % to the value
3)
Standardized Normal
• Z-Score: # of standard deviations x is from
the mean
• Z~N(0,1)
Inverse Normal
• When to use???
• How to do I use a calculator…?
• P(X<k) vs P(X>k)
Inverse Normal
• Use when given a probability and are
asked to calculate corresponding
measurement
• How to do I use a calculator…?
• P(X<k) vs P(X>k)
– Calculator gives Probability for area left
(meaning < ) of k
Example 1
• A university professor determines that no
more than 80% of this year’s History
candidates should pass the final examination.
The examination results were approximately
normally distributed with mean 62 and
standard deviation of 13. Find the lowest
score necessary to pass the exam.
• Draw the normal curve that illustrates the
situation
Example 2
• Seth is studying O-Chem and Economics. He sits
for the mid-year exams in each subject. His OChem mark is 56% and the class mean and
standard deviation are 50.2% and 15.8%
respectively. In Economics he is told that the
class mean and standard deviation are 58.7% and
18.7% respectively. What percentage does Seth
need to score in Economics, to have an equivalent
result to his O-Chem mark?
• Draw the normal curve that illustrates the situation
Poisson Distribution
• What is it?
• What is it for?
• Equation? Variables?
• Mean and Variance?
Poisson Distribution Answers
1.
Most widely used and applied distribution to real world situations. Allows
you to count the number of occurrences over a period or range (not
necessarily time).
2.
a)
b)
3.
4.
5.
6.
Answers vary. Examples?
See Last Class
See Last Class
a)
b)
7.
8.
9.
10.
Large # of potential emailers each w/ small probability of sending email
Area of land able to get
earthquakes, small probability of
earthquake at any
given moment
22.3%
6.564%
22.38%
35
9.88%
See Handout for remaining solutions
Distributions
Normal
Binomial
Poisson
Review
Events A and B are such that P(A) = 0.3 and
P(B) = 0.4.
(a) Find the value of P(A B) when
(i) A and B are mutually exclusive;
(ii)A and B are independent.
(b) Given that P(A B) = 0.6, find P(A | B).
Review
The fish in a lake have weights that are normally
distributed with a mean of 1.3 kg and a standard
deviation of 0.2 kg.
(a) Determine the probability that a fish that is caught
weighs less than 1.4 kg.
(b) John catches 6 fish. Calculate the probability that
at least 4 of the fish weigh more than 1.4 kg.
(c) Determine the probability that a fish that is caught
weighs less than 1 kg, given that it weighs less than
1.4 kg.
Review
• Find the probability of getting a pair on
your first roll in Yahtzee
Review
The ten numbers x1, x2, ..., x10 have a mean
of 10 and a standard deviation of 3.
Find the value of
Review
• A biased coin is weighted such that the
probability of obtaining a head is . The
coin is tossed 6 times and X denotes the
number of heads observed. Find the value
of the ratio
Review
After being sprayed with a weedkiller, the survival time of
weeds in a field is normally distributed with a mean of 15
days.
(a) If the probability of survival after 21 days is 0.2, find
the standard deviation of the survival time.
When another field is sprayed, the survival time of weeds
is normally distributed with a mean of 18 days.
(b) If the standard deviation of the survival time is
unchanged, find the probability of survival after 21 days.
Review
After a shop opens at 09:00 the number of customers arriving in any interval of
duration t minutes follows a Poisson distribution with mean .
(a)
(i) Find the probability that exactly five customers arrive
before
10:00.
(ii) Given that exactly five customers arrive before 10:00,
find
the probability that exactly two customers
arrive before
09:30.
(b) Let the second customer arrive at T minutes after 09:00.
(i) Show that, for t > 0,
P(T > t) =
(ii) Hence find in simplified form the probability density
function of T.
(iii) Evaluate E(T).
(You may assume that, for n + and a > 0, .)
Journal
Explain the purpose that the r! serves in the
Combination formula.