Statistical Inference: Section 1&2

Question 1

Define a random sample.(2)

Answer 1

all members of the population have the same chance of being included in the sample
all combinations of, say,nmembers have the same chance of being included in the sample.

Question 2

What does i.i.d stand for?(1)

Answer 2

Independent and identically distributed.

Question 3

Name common continuous distributions.(4)

Answer 3

Normal distribution
Standard normal distribution (special type of normal where mean=0,var=1)
Exponential
Uniform distribution.

Question 4

Name common discrete distributions. Difference between these?(3)

Answer 4

Binomial

Poisson (main difference to binomial is there is no upper limit on this eg “out of” like in binomial)

Question 5

What is a Bernoulli trial?(1)

Answer 5

Only 2 outcomes to the experiment, has binomial distribution if n fixed (no of experiments), constant of p probability and independent trials.

Question 6

How would you generate a random sample of 10 from N(100, 15^2) on R?(1)
*note N=normally distributed,
The mean?
Variance?(3)

Answer 6

1)normal_sample_1 = rnorm(10,100,15)
generates the sample and places it under “normal_sample_1”.
2)mean_1 = mean(normal_sample_1), storing it as “mean_1”
3)var_1 = var(normal_sample_1), storing as “var_1”.

Question 7

What is simulation?(1)

Answer 7

Using samples of a known population to test the means of the estimates, hence when it comes to unknown populations we have greater confidence in these estimates.

Question 8

How would you generate a poisson sample of 20 from Po(4) on R?(1)
The mean?
Variance?(3)

Answer 8

> poisson_sample = rpois(20,4)
poisson_mean = mean(poisson_sample)
poisson_var = var(poisson_sample).

Question 9

Binomial distribution where the number of trials is equal to 100, and the success probability is equal to 0.5. Sample mean? (2)
Note this is equivalent to tossing a fair coin 100 times and counting the number of heads.

Answer 9

> binomial_sample = rbinom(1,100,0.5)

> binomial_sample[1] 46

Question 10

What is exploratory data analysis?(1)

Answer 10

If we want to try and assess how well a particular probability distribution might work as a model for some data, we need to have a look at the data.

Question 11

What is a 5 number summary?How do you find this in R?What about if you wanted the mean too?(3)

Answer 11

Min, LQ, Median, UQ, Max
quantile(x) gives five number summary
summary(x) gives this and the mean.

Question 12

Define the sample mean.(1)

Answer 12

For a sample consisting of n observations x1,x2,…,xn, the sample mean ̄x is defined as the arithmetic mean of the observations, i.e. ̄x=1/n*∑Xi–>n i=1

Question 13

What does W denote?(1)

Answer 13

Theθj, j= 1,…,pwill belong to a set of valuesW, called the parameter space, and soXis a member of a family of distributions.

Question 14

Define sample variance.(1)

Answer 14

For a sample consisting of n observations x1,x2,…,xn, the sample variance s^2 is defined as s^2=1/(n−1)∑i=1 to n for (xi− ̄x)^2.

Question 15

Difference between estimator and estimate?(1)

Answer 15

This process, which we can repeat, is our estimator, and any particular sample gives us an estimate.

Estimators are conclusions drawn about the population from a sample ie a sample mean would be an estimator for the population mean.

Question 16

What is a sample statistic?What is special about these?(2)

Answer 16

Any particular function defined on the random sample,

note that a sample statistic is a function of random variables, and so itself is a random variable with its own distribution, an expectation and a variance.

Question 17

Important properties of estimators.(3)

Answer 17

1)Bias, represented as E[theta-hat]-theta
theta-hat=estimate, smaller bias=better, 0 bias=unbiased estimator
2)Variance, smaller the better, if optimally small (achieves a theoretically lower limit) is known as “efficient”
3)BEST MEASURE- as involves bias and variance-Mean square error (MSE) is E[(theta-hat - theta)^2] combines both bias and variance, when not possible to minimise both bias and variance simultaneously then minimising MSE is a good compromise.

Question 18

How to work out probabilities from a normal distribution using R?(1)

Answer 18

Use the pnorm function
pnorm(x, mean=, sd=, lower.tail=TRUE/FALSE)

true means< or equal
false means> .

Question 19

How would you do the following question in r:

What proportion of trusts have N1 more than 20% higher than E1, ie more than 1.2×E1? (1)

Answer 19

length(which(hospitals$N1>1.2*hospitals$E1))

Question 20

How would you do the following question in r:
Suppose trusts are independent and a further sample of 10 trusts is selected randomly.
Let W be the number of new trusts that have N1 more than 20% higher than E1. Use
your answer to the previous question (0.16) to estimate
(a) Pr(W = 0), [3dp] ( 10 marks)
(b) Var(W).

Answer 20

Binomial distribution as INDEPENDENT and TWO outcomes

a) pbinom(0,10,0.16)
b) Variance=np(1-p).

Question 21

What is the POPULATION expectation of an exponential random variable?(1)

Answer 21

1/lambda

Question 22

What would be the parameter space for the normal distribution?For exponential?(2)

Answer 22

{(μ,σ2) :μ∈R,σ >0}.

W={λ:λ >0}.

Question 23

What is a parameter?(1)

Answer 23

A numerical summary of a POPULATION/distribution, usually unknown eg if total cars in US was population, total number of red cars would be parameter as unknown and difficult to know.

Question 24

What is a statistic?(1)

Answer 24

Summary of data/sample function of SAMPLE

Question 25

What is a sample statistic?(1)

Answer 25

Any function of the of the random SAMPLE
Note: A sample statistic is a function of random variables, and so itself is a random variable with its own distribution, an expectation and a variance.

Question 26

Trade off between bias and variance, give example.(1)

Answer 26

Can trade of bias for a smaller variance, think of 2 archers one with no bias but lots of variance and one with a little bias but no variance, the second archer would be better.

Question 27

For the following independent distributions X~Po(lambda) Y~Po(2lambda), classify the following as statistics, parameters or neither:

a) X+2Y
b) lambdahat=function of X&Y
c) lambdahat/lambda
d) Pr

Answer 27

a) A function of random variables so statistic
b) Statistic
c) Neither, stat and parameter
d) Parameter.

Question 28

Poisson distribution expected value, variance, parameter.(1)

Answer 28

All the same.

Eg X~Po(lambda) then E[X]=lambda=Var[X]

Question 29

Variance of the sum of INDEPENDENT random variables

Var(aX+bY)=

Answer 29

a^2Var(X)+b^2Var(Y).

Question 30

If you have an unbiased sample what is the MSE equal to?(1)

Answer 30

The variance.

Question 31

For the following independent distributions X~Po(lambda) Y~Po(2lambda) what estimator would you recommend?

See week 2 online for full example.

Answer 31

Ideally want unbiased estimator so w1X+w2Y=lambda (the target)
Therefore MSE=variance
plug these in and get quadratic for w2 solve for minimum and get solutions.

Question 32

What is the Central Limit Theorem?Importance?(4)

Answer 32

Xn−→dN(μ,σ2/n) as n→∞.
Ie, for set of iid variables X1,X2,X3… there is a mean mu and variance sigma^2, Xbar defined as mean of first n terms (note this means Xbar is its own random variable woth distribution)

Central limit basically states that irrespective of the distribution, for n increasing it will tend to a normal distribution with variance sigma^2/n and mean of the mu

OR
If you standardise (ie subtract mean divide by sd) you get normal distribution now as n tends to infinity this converges to N~(0,1) ie normal with 0 mean and variance 1.

Can approximate, if exam q doesn’t mention distribution use this!

Question 33

In the CLT what does the little d on the arrow mean in the formula?(1)

Answer 33

Means convergence of the distribution ie is changing from previous to the normal.
Also means you have a random limit

Question 34

Large sample (asymptotic) properties of estimators.(3)

Answer 34

1) Asymtotic unbiasedness (E[ˆθ]→θ as n→ ∞) Note that if ˆθ is unbiased for all n, thenE[ˆθ] =θ, and it automatically follows that ˆθ is asymptotically unbiased
2) Consistency: met by 2 things:
a) Need to be asymptotic unbiased
b) V ar[ˆθ]→0 as n→∞
3)Asymptotic efficiency-said to be this if:
Variance converges to theoretical lower limit as n tends to infinity ie (actual variance / lower limit)→1 as n→∞.

Question 35

Incorrect question on assignment correction:

Suppose Y is an exponential random variable with expected value equal to the median of |N2 - E2|. What is the variance ofY?[2dp] ( 10 marks)

Answer 35

Answer correction:

The Exp(λ) distribution has expected value 1/λ, variance 1/λ2 and cumulative distribution function

((F(x) = Pr(X \leq x) = \left{ \begin{array}{ll} 0 & \hspace*{0.5cm} x<0\ 1- e^{-\lambda x} & \end{array}))

> lambda=1/median(tem)
round(1/lambda^2,2)
[1] 14.44

Question 36

When do you refer to pdf and when for pmf?(1)

Answer 36

R has functions for the uniform, exponential and normal distributions, where we now refer to the
probability density function, rather than probability mass function for discrete distributions like poisson and binomial.

Question 37

When calculating confidence intervals in r how do you do it?(1)

Answer 37

For normal random use qnorm function and use probability with half the amount wanted ie
for 95 confidence interval use qnorm(0.975) note 2.5 is half the 5 wanted
for 90% CI use qnorm(0.95)
for 99% CI use qnorm(0.995)

Question 38

If asked to calculate a sample size given a required confidence interval, how would you do this?(1)

Answer 38

Make it the difference bertween the confidence intervals, winds up being 2* the critical value*sd including n and you rearrange to find n, rounding up to get a smaller interval.

Question 39

Variance for sample statistic mean and regular?(1)

Answer 39

sigma^2 is regular

sigma^2/n is the variance for the mean of a sample.

Question 40

What is the pdf of an exponential random variable?The cdf?(2)

Answer 40

fx(x)=lambdae^(-lambdax)

cdf=Fx(x)=1-e^(-lambda*x)