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PROBABILITY AND
STATISTICS
WEEK 8
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Skewness
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Pearson's skewness coefficient
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Example
• It’s been understood that, in a hosptial patients’
average hospital stay is 28, median is 25 and
mode is 23 (days). And the standard deviation
calculated as 4,2.
• Define the skewness type, find the pearson
coefficient and interpret it.
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Joint Probability Distributions
• Discrete Random Variables (two and multiple)
• Continious Random Variables (two and multiple)
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Two Discrete Random Variables
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Example
A large insurance agency services a number of customers who have purchased
both a homeowner’s policy and an automobile policy from the agency. For
each type of policy, a deductible amount must be specified. For an automobile
policy, the choices are $100 and $250, whereas for a homeowner’s policy, the
choices are 0, $100, and $200.
Suppose an individual with both types of policy is selected at random from the
agency’s files. Let X the deductible amount on the auto policy and Y the
deductible amount on the homeowner’s policy.
Suppose the joint pmf is given in the accompanying joint probability table:
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Example
• P(Y ≥100)=?
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Marginal Probability Mass Function
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Example
• Let us obtain the marginal pmf of X evaluated
at, x=100 and x=250.
• Let us obtain the marginal pmf of Y.
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Remember
• We already know!
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Conditional Probability Mass Function
• Recall: P(A/B)=P(AB)/P(B)
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Example
Opinion about school (X) and teachers (Y) surveyed for students as follows:
School Scores
Teachers Scores
1
2
3
1
30
10
20
2
50
30
10
3
10
10
30
a) Determine the joint probability distribution of X and Y.
b) Find the probability student give 1 point for school and 2 or above point for
teachers?
c) Find the marginal probability distribution, mean and variance of school scores.
d) Determine the conditional probability distribution of teacher scores given that
school score is 2. And find the (conditional) mean and variance of Y given X=2.
e) Check if school opinions and teacher opinions are independent.
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Two Continuous Random Variables
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Example
A bank operates A and B facilities. On a randomly selected day, let X
the proportion of time that A in use and Y the proportion of time that
the B in use. Then the set of possible values for (X, Y) is the
rectangle
D = {(x, y): 0≤ x ≤ 1, 0 ≤ y ≤ 1}.
Suppose the joint pdf of (X, Y) is given by,
a) Verify that this is a legitimate pdf.
b) Find the probability that neither facility is busy more than onequarter of the time.
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Marginal Probability Density Function
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Example
• Find the marginal pdf of X and Y for previous
question.
• Find P(1/4≤Y ≤3/4)=?
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Example
A nut company markets cans of deluxe mixed nuts containing almonds,
cashews, and peanuts. Suppose the net weight of each can is exactly 1
lb, but the weight contribution of each type of nut is random. Because
the three weights sum to 1, a joint probability model for any two gives
all necessary information about the weight of the third type. Let X = the
weight of almonds in a selected can and Y = the weight of cashews.
• Verify that this is a legitimate pdf.
• To compute the probability that the two types of nuts together make up at
most 50% of the can,
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Conditional Probability Mass Function
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Example
The sizes of college books are under study in terms of width (X) and
height (Y). Suppose that the joint p.d.f of X and Y is modeled as
f (x,y)=cxy, 2<x<5 and x+ 1 <y<x+3 (unit:inch)
•Determine the value of c.
•Determine the marginal probability distribution of X. Find also the
mean and variance of X.
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Covariance
• Covariance and correlation between x and y?
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Covariance
Shortcut formula:
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Correlation
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Example
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