Review DS GA 1002 Statistical and Mathematical Models - - PowerPoint PPT Presentation

Review

DS GA 1002 Statistical and Mathematical Models

http://www.cims.nyu.edu/~cfgranda/pages/DSGA1002_fall16 Carlos Fernandez-Granda

Probability and statistics

◮ Probability:

Framework for dealing with uncertainty

◮ Statistics:

Framework for extracting information from data making probabilistic assumptions

Probability

◮ Probability basics: probability spaces, conditional probability,

independence, conditional independence

◮ Random variables: pmf, cdf, pdf, important distributions, functions of

random variables

◮ Multivariate random variables: joint pmf, joint cdf, joint pdf, marginal

distributions, conditional distributions, independence, joint distribution

• f discrete/continuous random variables

Probability

◮ Expectation: definition, mean, median, variance, Markov and

Chebyshev inequalities, covariance, correlation coefficient, covariance matrix, conditional expectation

◮ Random processes: definition, mean, autocovariance, important

processes (iid, Gaussian, Poisson, random walk), Markov chains

◮ Convergence: types of convergence, law of large numbers, central limit

theorem, convergence of Markov chains

◮ Simulation: motivation, inverse-transform sampling, rejection

sampling, Markov-chain Monte Carlo

Statistics

◮ Descriptive statistics: histogram, empirical mean/variance, order

statistics, empirical covariance, principal component analysis

◮ Statistical estimation: frequentist perspective, mean square error,

consistency, confidence intervals

◮ Learning models: method of moments, maximum likelihood, empirical

cdf, kernel density estimation

Statistics

◮ Hypothesis testing: definitions (null/alternative hypothesis, Type I/II

errors), significance level, power, p value, parametric testing, power function, likelihood-ratio test, permutation test, multiple testing, Bonferroni’s method

◮ Bayesian statistics: prior, likelihood, posterior, posterior mean/mode ◮ Linear regression: linear models, least squares, geometric

interpretation, probabilistic interpretation, overfitting

Random walk with a drift

We define the random walk X as the discrete-state discrete-time random process

• X (0) := 0,
• X (i) :=

X (i − 1) + S (i) + 1, i = 1, 2, . . . where

• S (i) =
• +1

with probability 1

2,

−1 with probability 1

2,

is an iid sequence of steps

Random walk with a drift

What is the mean of this random process? E

• X (i)

Random walk with a drift

What is the mean of this random process? E

• X (i)
• = E

 

i

• j=1
• S (j) + 1
• 



Random walk with a drift

What is the mean of this random process? E

• X (i)
• = E

 

i

• j=1
• S (j) + 1
• 

 =

i

• j=1

E

• S (j)
• + n

Random walk with a drift

What is the mean of this random process? E

• X (i)
• = E

 

i

• j=1
• S (j) + 1
• 

 =

i

• j=1

E

• S (j)
• + n

= i

Random walk with a drift

What is the autocovariance? Use the fact that the autocovariance of the random walk without drift W that we studied in the lecture notes is R

W (i, j) = min {i, j}

Random walk with a drift

• X (i)

Random walk with a drift

• X (i) =

W (i) + i

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

R

X (i, j)

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

R

X (i, j) := E

• X (i)

X (j)

• − E
• X (i)
• E
• X (j)

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

R

X (i, j) := E

• X (i)

X (j)

• − E
• X (i)
• E
• X (j)
• = E
• W (i) + i
• W (j) + j
• − E
• W (i) + i
• E
• W (j) + j

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

R

X (i, j) := E

• X (i)

X (j)

• − E
• X (i)
• E
• X (j)
• = E
• W (i) + i
• W (j) + j
• − E
• W (i) + i
• E
• W (j) + j
• = E
• W (i)

W (j)

• + iE
• W (j)
• + jE
• W (i)
• + ij

− iE

• W (j)
• − jE
• W (i)
• − ij

Random walk with a drift

• X (i) =

W (i) + i E

• W (i)
• = 0

R

X (i, j) := E

• X (i)

X (j)

• − E
• X (i)
• E
• X (j)
• = E
• W (i) + i
• W (j) + j
• − E
• W (i) + i
• E
• W (j) + j
• = E
• W (i)

W (j)

• + iE
• W (j)
• + jE
• W (i)
• + ij

− iE

• W (j)
• − jE
• W (i)
• − ij

= min {i, j}

Random walk with a drift

Compute the first-order pmf of X (i). Recall that the first-order pmf of the random walk W equals p

W (i) (x) =

i

i+x 2

1

2i

if i + x is even and −i ≤ x ≤ i

• therwise

Random walk with a drift

p

X(i) (x)

Random walk with a drift

p

X(i) (x) = P

• X (i) = x

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i
• = p

W (i) (x − 1)

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i
• = p

W (i) (x − 1)

=

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i
• = p

W (i) (x − 1)

= i

x 2

1

2i

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i
• = p

W (i) (x − 1)

= i

x 2

1

2i

if x is even and 0 ≤ x ≤ 2i

Random walk with a drift

p

X(i) (x) = P

• X (i) = x
• = P
• W (i) = x − i
• = p

W (i) (x − 1)

= i

x 2

1

2i

if x is even and 0 ≤ x ≤ 2i

• therwise

Random walk with a drift

Does the process satisfy the Markov condition? p

X(i+1) | X(1), X(2),..., X(i) (xi+1 | x1, x2, . . . , xi) = p X(i+1) | X(i) (xi+1|xi)

Random walk with a drift

p

X(i+1) | X(1), X(2),..., X(i) (xi+1 | x1, x2, . . . , xi)

Random walk with a drift

p

X(i+1) | X(1), X(2),..., X(i) (xi+1 | x1, x2, . . . , xi)

= P

• xi +

S (i + 1) + 1 = xi+1

Random walk with a drift

p

X(i+1) | X(1), X(2),..., X(i) (xi+1 | x1, x2, . . . , xi)

= P

• xi +

S (i + 1) + 1 = xi+1

• = p

X(i+1) | X(i) (xi+1|xi)

Random walk with a drift

We observe that X (10) = 16 and X (20) = 30. What is the best estimator for X (21) in terms of probability of error?

Random walk with a drift

p

X(21) | X(10), X(20) (x|16, 30)

Random walk with a drift

p

X(21) | X(10), X(20) (x|16, 30) = p X(21) | X(20) (x|30)

Random walk with a drift

p

X(21) | X(10), X(20) (x|16, 30) = p X(21) | X(20) (x|30)

=     

1 2

if x = 32

1 2

if x = 30

• therwise

Markov chain

Consider a Markov chain X with transition matrix T

X :=

• a

1 1 − a

• ,

where a is a constant between 0 and 1. We label the two states 0 and 1. The transition matrix T

X has two eigenvectors

• q1 :=
• 1

1−a

1

• ,
• q2 :=

1 −1

• The corresponding eigenvalues are λ1 := 1 and λ2 := a − 1

Markov chain

For what values of a is the Markov chain irreducible?

Markov chain

For what values of a is the Markov chain periodic?

Markov chain

Express the stationary distribution of X in terms of a

• pstat

Markov chain

Express the stationary distribution of X in terms of a

• pstat =

1 ( q1)1 + ( q1)2

• q1

Markov chain

Express the stationary distribution of X in terms of a

• pstat =

1 ( q1)1 + ( q1)2

• q1

= 1 2 − a

• 1

1 − a

Markov chain

Does the Markov chain always converge in probability for all values of a? Justify that this is the case or provide a counterexample.

Markov chain

Express the conditional pmf of X (i) conditioned on X (1) = 0 as a function

• f a and i. (Hint: Computing

q1 + q2 could be a helpful first step.) Evaluate the expression at a = 0 and a = 1. Does the result make sense?

Markov chain

We have

• q1 +

q2

Markov chain

We have

• q1 +

q2 =

• 1

1−a

1

• +

1 −1

Markov chain

We have

• q1 +

q2 =

• 1

1−a

1

• +

1 −1

• =

2−a

1−a

Markov chain

We have

• q1 +

q2 =

• 1

1−a

1

• +

1 −1

• =

2−a

1−a

• p

X(0)

Markov chain

We have

• q1 +

q2 =

• 1

1−a

1

• +

1 −1

• =

2−a

1−a

• p

X(0) =

1

Markov chain

We have

• q1 +

q2 =

• 1

1−a

1

• +

1 −1

• =

2−a

1−a

• p

X(0) =

1

• = 1 − a

2 − a ( q1 + q2)

Markov chain

• p

X(i)

Markov chain

• p

X(i) = T i

• X

p

X(0)

Markov chain

• p

X(i) = T i

• X

p

X(0)

= T i

• X

1 − a 2 − a ( q1 + q2)

Markov chain

• p

X(i) = T i

• X

p

X(0)

= T i

• X

1 − a 2 − a ( q1 + q2) = 1 − a 2 − a

• λi

1

q1 + λi

2

q2

Markov chain

• p

X(i) = T i

• X

p

X(0)

= T i

• X

1 − a 2 − a ( q1 + q2) = 1 − a 2 − a

• λi

1

q1 + λi

2

q2

• = 1 − a

2 − a

• 1

1−a

1

• + (a − 1)i

1 −1

Markov chain

• p

X(i) = T i

• X

p

X(0)

= T i

• X

1 − a 2 − a ( q1 + q2) = 1 − a 2 − a

• λi

1

q1 + λi

2

q2

• = 1 − a

2 − a

• 1

1−a

1

• + (a − 1)i

1 −1

• =

1 2 − a

• 1 − (a − 1)i+1

(1 − a)

• 1 − (a − 1)i

Markov chain

For a = 1 we have

• p

X(i) =

1

Markov chain

For a = 0 we have

• p

X(i) = 1

2

• 1 − (−1)i+1

1 − (−1)i

Markov chain

For a = 0 we have

• p

X(i) = 1

2

• 1 − (−1)i+1

1 − (−1)i

• =

          

• 1
• if i is odd,
• 1
• if i is even.

Sampling from multivariate distributions

We are interested in generating samples from the joint distribution of two random variables X and Y . If we generate a sample x according to the pdf fX and a sample y according to the pdf fY , are these samples a realization

• f the joint distribution of X and Y ? Explain your answer with a simple

example.

Sampling from multivariate distributions

Now, assume that X is discrete and Y is continuous. Propose a method to generate a sample from the joint distribution using the pmf of X and the conditional cdf of Y given X using two independent samples from a distribution that is uniform between 0 and 1. Assume that the conditional cdf is invertible.

Sampling from multivariate distributions

• 1. Obtain two independent samples u1 and u2 from the uniform

distribution.

Sampling from multivariate distributions

• 1. Obtain two independent samples u1 and u2 from the uniform

distribution.

• 2. Set x to equal the smallest value a such that pX (a) = 0 and

u1 ≤ FX (a).

Sampling from multivariate distributions

• 1. Obtain two independent samples u1 and u2 from the uniform

distribution.

• 2. Set x to equal the smallest value a such that pX (a) = 0 and

u1 ≤ FX (a).

• 3. Define

Fx (·) := FY | X (· | x) Set y := F −1

x

(u2)

Sampling from multivariate distributions

Explain how to generate samples from a random variable with pdf fW (w) = 0.1 λ1 exp (−λ1w) + 0.9 λ2 exp (−λ2w) , w ≥ 0, where λ1 and λ2 are positive constants, using two iid uniform samples between 0 and 1.

Sampling from multivariate distributions

Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ1 and if X = 1 then Y is exponential with parameter λ2

Sampling from multivariate distributions

Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ1 and if X = 1 then Y is exponential with parameter λ2 The marginal distribution of Y is fY (w) = pX (0) fY | X (w | 0) + pX (1) fY | X (w | 1)

Sampling from multivariate distributions

Let us define a Bernoulli random variable X with parameter 0.9, such that if X = 0 then Y is exponential with parameter λ1 and if X = 1 then Y is exponential with parameter λ2 The marginal distribution of Y is fY (w) = pX (0) fY | X (w | 0) + pX (1) fY | X (w | 1) = 0.1 λ1 exp (−λ1w) + 0.9 λ2 exp (−λ2w)

Sampling from multivariate distributions

• 1. We obtain two independent samples u1 and u2 from the uniform

distribution.

Sampling from multivariate distributions

• 1. We obtain two independent samples u1 and u2 from the uniform

distribution.

• 2. If u1 ≤ 0.1 we set

w := 1 λ1 log

• 1

1 − u2

• therwise we set

w := 1 λ2 log

• 1

1 − u2

Convergence

Let U be a random variable uniformly distributed between 0 and 1. If we define the discrete random process X

• X (i) = U for all i,

does X converge to 1 − U in probability?

Convergence

Does X converge to 1 − U in distribution?

Convergence

You draw some iid samples x1, x2, . . . from a Cauchy random variable. Will the empirical mean 1

n

n

i=1 xi converge in probability as n grows large?

Explain why briefly and if the answer is yes state what it converges to.

Convergence

You draw m iid samples x1, x2, . . . , xm from a Cauchy random variable. Then you draw iid samples y1, y2, . . . uniformly from {x1, x2, . . . , xm} (each yi is equal to each element of {x1, x2, . . . , xm} with probability 1/m). Will the empirical mean 1

n

n

i=1 yi converge in probability as n grows large?

Explain why very briefly and if the answer is yes state what it converges to.

Earthquake

We are interested in learning a model for the occurrence of earthquakes. We decide to model the time between earthquakes as an exponential random variable with parameter λ. Compute the maximum-likelihood estimate of λ given t1, t2, . . . , tn, which are interarrival times for past earthquakes. Assume that the data are iid.

Earthquake

L (λ)

Earthquake

L (λ) := f

T(1),..., T(n) (t1, . . . , tn)

Earthquake

L (λ) := f

T(1),..., T(n) (t1, . . . , tn)

=

n

• i=1

λ exp (−λti)

Earthquake

L (λ) := f

T(1),..., T(n) (t1, . . . , tn)

=

n

• i=1

λ exp (−λti) = λn exp

• −λ

n

• i=1

ti

Earthquake

L (λ) := f

T(1),..., T(n) (t1, . . . , tn)

=

n

• i=1

λ exp (−λti) = λn exp

• −λ

n

• i=1

ti

• log L (λ)

Earthquake

L (λ) := f

T(1),..., T(n) (t1, . . . , tn)

=

n

• i=1

λ exp (−λti) = λn exp

• −λ

n

• i=1

ti

• log L (λ) = n log λ − λ

n

• i=1

ti

Earthquake

d log Lt1,...,tn (λ) dλ

Earthquake

d log Lt1,...,tn (λ) dλ = n λ −

n

• i=1

ti

Earthquake

d log Lt1,...,tn (λ) dλ = n λ −

n

• i=1

ti d2 log Lt1,...,tn (λ) dλ2

Earthquake

d log Lt1,...,tn (λ) dλ = n λ −

n

• i=1

ti d2 log Lt1,...,tn (λ) dλ2 = −n λ2

Earthquake

d log Lt1,...,tn (λ) dλ = n λ −

n

• i=1

ti d2 log Lt1,...,tn (λ) dλ2 = −n λ2 λML

Earthquake

d log Lt1,...,tn (λ) dλ = n λ −

n

• i=1

ti d2 log Lt1,...,tn (λ) dλ2 = −n λ2 λML = 1

1 n

n

i=1 ti

Earthquake

Find an approximate 0.95 confidence interval based on the central limit theorem for the value of λ. Assume that you know a bound b on the standard deviation (i.e. the variance of the exponential 1/λ2 is bounded by b2) and express your answer using the Q function. (Hint: Express the ML estimate in terms of the empirical mean.) (See solutions.)

Earthquake

What is the posterior distribution of the parameter Λ if we model it as a random variable with a uniform distribution between 0 and u? Express your answer in terms of the sum n

i=1 ti, u and the marginal pdf of the data

evaluated at t1, t2, . . . , tn c := f

T(1),..., T(n) (t1, . . . , tn) .

Earthquake

fΛ |

T(1),..., T(n) (λ | t1, . . . , tn)

Earthquake

fΛ |

T(1),..., T(n) (λ | t1, . . . , tn) = fΛ (λ) λn exp (−λ n i=1 ti)

f

T(1),..., T(n) (t1, . . . , tn)

Earthquake

fΛ |

T(1),..., T(n) (λ | t1, . . . , tn) = fΛ (λ) λn exp (−λ n i=1 ti)

f

T(1),..., T(n) (t1, . . . , tn)

= 1 u c λn exp

• −λ

n

• i=1

ti

Earthquake

fΛ |

T(1),..., T(n) (λ | t1, . . . , tn) = fΛ (λ) λn exp (−λ n i=1 ti)

f

T(1),..., T(n) (t1, . . . , tn)

= 1 u c λn exp

• −λ

n

• i=1

ti

• for 0 ≤ λ ≤ u and zero otherwise

Earthquake

λ fΛ |

T(1),..., T(n) (λ | t1, . . . , tn)

Earthquake

Explain how you would use the answer in the previous question to construct a confidence interval for the parameter

Chad

You hate a coworker and want to predict when he is in the office from the temperature. Chad 61 65 59 61 61 65 61 63 63 59 No Chad 68 70 68 64 64

• You model his presence using a random variable C which is equal to 1 if he

is there and 0 if he is not. Estimate pC.

Chad

The empirical pmf is pC (0) = 5 15 = 1 3, pC (1) = 10 15 = 2 3.

Chad

You model the temperature using a random variable T. Sketch the kernel density estimator of the conditional distribution of T given C using a rectangular kernel with width equal to 2.

Chad

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 0.00 0.05 0.10 0.15 0.20 fT|C(t|0) fT|C(t|1)

Chad

If T = 68◦ what is the ML estimate of C?

Chad

If T = 68◦ what is the ML estimate of C? fT|C (68|0) = 0.2 fT|C (68|1) = 0

Chad

If T = 64◦ what is the MAP estimate of C?

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64)

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1)

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1) =

1 30.2 1 30.2 + 2 30.1

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1) =

1 30.2 1 30.2 + 2 30.1

= 1 2

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1) =

1 30.2 1 30.2 + 2 30.1

= 1 2 pC|T (1|64)

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1) =

1 30.2 1 30.2 + 2 30.1

= 1 2 pC|T (1|64) = 1 − pC|T (0|64)

Chad

If T = 64◦ what is the MAP estimate of C? pC|T (0|64) = pC (0) fT|C (64|0) pC (0) fT|C (64|0) + pC (1) fT|C (64|1) =

1 30.2 1 30.2 + 2 30.1

= 1 2 pC|T (1|64) = 1 − pC|T (0|64) = 1 2

Chad

What happens if the temperature is 57◦? Explain how using parametric estimation may alleviate this problem.

3-point shooting

The New York Knicks hire you as a data analyst. Your first task is to come up with a way to determine whether a 3-point shooter is any good. You will use the following graph of the function g (θ, n) = θn.

0.5 0.6 0.7 0.8 0.9 1.0

θ

0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 0.005

g(θ,n)

n = 4 n = 9 n = 14 n = 19 n = 24

3-point shooting

• 1. Interpret g (θ, n).

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

• 6. You apply the test on 10 players. You adapt the threshold applying

Bonferroni’s method. What is the new threshold?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

• 6. You apply the test on 10 players. You adapt the threshold applying

Bonferroni’s method. What is the new threshold? n = 24

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

• 6. You apply the test on 10 players. You adapt the threshold applying

Bonferroni’s method. What is the new threshold? n = 24

• 7. With the correction, what is the probability that you do not declare a

player who has a shooting percentage of 90% as a good shooter?

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

• 6. You apply the test on 10 players. You adapt the threshold applying

Bonferroni’s method. What is the new threshold? n = 24

• 7. With the correction, what is the probability that you do not declare a

player who has a shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.92

3-point shooting

• 1. Interpret g (θ, n).
• 2. The coach tells you: I want to make sure that the guy has a shooting

percentage over 80%. What is your null hypothesis?

• 3. What number of shots does a player need to make in a row for you to

reject the null hypothesis with a confidence level of 5%? 14

• 4. A player makes 9 shots in a row. What is the corresponding p value?

Do you declare him as a good shooter? ≈ 0.14

• 5. What is the probability that you do not declare a player who has a

shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.76

• 6. You apply the test on 10 players. You adapt the threshold applying

Bonferroni’s method. What is the new threshold? n = 24

• 7. With the correction, what is the probability that you do not declare a

player who has a shooting percentage of 90% as a good shooter? 1 − g (0.9, 14) ≈ 0.92

• 8. What is the advantage of adapting the threshold? What is the

disadvantage?