Transcript Lab 5 Discrete Random Variables and Linear Combinations

Lab 6 The Binomial Distribution
Spring 2013
Lab Exam1
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Coverage: Chapter2~Chapter6.
Four questions in 50 minutes.
Sample Exams, Lab Activities
Trick: No tricky questions but need to be fast.
Practice and Practice.
Come to class ten minutes earlier, you need to
check if Excel works and download the data set.
Review-Ch2 Graph
• Distinguish data type and graphs used to analyze
each type: Numerical or Categorical
• How to draw the histogram?
– Set bin range: equal interval, be careful with upper
bound
– Data analysis Histogram
• How to read the histogram?
– Summary table shows data frequency for each interval
• Ogive can give you the cumulative frequency.
How to read it? Summary table.
Review-Ch3 Summary Statistics
• Three dimensions of distribution: Center, Dispersion,
Shape. Know measures for these three dimensions.
• If your data set is categorical use mode
• If data set is pop. or sample depends on question
– Interest on the data set only treat data set as population.
– Interest on the whole pop treat data set as sample.
• Summary statistics in Data Analysis gives you sample
variance.
• Coefficient of Variance can compare the relative
dispersion.
• Shape: Skewness coefficient or mean-median.
• Interpret those measures.
Review-Ch4 Two Variables Analysis
• Data Analysis gives you population covariance
matrix and correlation matrix
• If you want to measure the strength of linear
relationship Use Correlation matrix
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• Covariance to Correlation and vice verse
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• How to draw a scatter plot and find the least
square line?
• How to interpret the least square line? Slope?
Intercept? Prediction?
XY
XY
X
Y
Review-Ch5 Linear Combinations
W=aX+bY
𝑬 𝑾 = 𝒂𝑬 𝑿 + 𝒃𝑬(𝒀)
𝑽 𝑾
= 𝒂𝟐 𝑽 𝑿 + 𝒃𝟐 𝑽 𝒀 + 𝟐𝒂𝒃𝑪𝒐𝒗(𝑿, 𝒀)
W=aX+bY+cZ
𝐸 𝑊 = 𝑎𝐸 𝑋 + 𝑏𝐸 𝑌 + 𝑐𝐸 𝑍
𝑉 𝑊
= 𝑎2 𝑉 𝑋 + 𝑏2 𝑉 𝑌 + 𝑐 2 𝑉 𝑍
+ 2𝑎𝑏 𝐶𝑜𝑣 𝑋, 𝑌 + 2𝑏𝑐 𝐶𝑜𝑣 𝑌, 𝑍
+ 2𝑐𝑎 𝐶𝑜𝑣(𝑍, 𝑋)
Cov(X,Y)=0 when X and Y are independent with each other.
Where are we?
• We learned how to describe distribution
• We will introduce five important distributions in
this class. They are:
Discrete Distribution
Binomial Distribution
X~ B(n, 𝜋)
Continuous Distribution
Uniform
Distribution
X~U(a,b)
𝑎+𝑏
E(X): 𝑛 ∗ 𝜋
E(X):
Var(X): 𝑛 ∗ 𝜋 ∗ (1 − 𝜋)
Var(X):
Shape: See next slide
Shape: Symmetric
2
(𝑏−𝑎)2
12
Normal
Distribution
X~N(𝜇, 𝜎)
Standard
Normal Dist
Z~N(0,1)
T-distribution
T~t(df)
E(X)=𝜇
E(X)= 0
E(X)= 0
Var(X)=𝜎 2
Var(X)= 1
Shape:symmetric
Shape:symmetric
Var(X)= depends
on d.f.
Shape:symmetric
Shape of Binomial Distribution
• If 𝑛 ∗ 𝜋 ≥ 5 and 𝑛 ∗ (1 − 𝜋) ≥ 5
 Binomial Distribution is Symmetric
• If either one of the above conditions does not
hold, then
 If 𝜋 > 0.5 left-skewed
 If 𝜋 = 0.5 Symmetric
 If 𝜋 < 0.5 Right-skewed
Excel’s BINOMDIST command
• In Excel, =BINOM.DIST(x, n, π, cumulative) is
used to calculate any binomial probabilities.
– X: number of successes (X ≥ 0, integer)
– n: number of trials (n ≥ 1, integer)
– π: probability of success (0 < π < 1)
– cumulative: logical argument to determine the
form of the function (0 or 1). “0” means “point
probability”, eg: P(X = x), and “1” means
“cumulative probability”, eg: P(X<= 3).
Cases
• Probability of X less than 6
P(X<6) = P(X≤5) = BINOMDIST(5, n, π, 1)
• Probability of X at most 6
P(X ≤ 6) = BINOMDIST(6, n, π, 1)
• Probability of X more than 6
P(X>6) = 1 - P(X≤6) = 1 - BINOMDIST(6, n, π, 1)
• Probability of X at least 6
P(X≥6) = 1 - P(X≤5) = 1 - BINOMDIST(5, n, π, 1)