Partial Derivatives Variance & Normal Distribution Statistics Worksheet

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1.Let the function f: R 2 → R be given by f (x, y) = x 3 - 3x - y 3 + 3y. (a) Find the partially derived of 1st and 2nd order. (b) In which direction does the function grow most in point (2,2)? (c) Find and classify the stationary points of the function. 2. Assume that a stochastic variable X is normally distributed with unknown mean µ and unknown variance σ 2 . You make 10 measurements on X and get the following result: 58,73,60,64,68,60,63,67,64,77 (a) Find a 95% confidence interval for µ. (b) Assume that you are told that X has known variance σ 2 = 36 = 6 2 . Find a 95% confidence interval for µ. Explain in words why it is naFortunately, this confidence interval is less than the interval you found above. (c) How many tests must be performed to ensure that the length of the the interval should be less than 1? (d) Is it possible to determine with certainty how many attempts must be perform to determine a confidence interval of length 1 if X has unknown variance? Base the answer. 3. In the development of a vaccine against Covid-19, researchers at a pharmaceutical company try out a test vaccine on a group of people. All the people in the group tested negative for Covid-19 when the experiment started. They divide the group of people involved in the experiment into 2 parts. One half (control group) receive placebo. After one month, 1% of the control group tested positive for Covid-19. The other half of the group receives the test vaccine. Let X be the stochastic variable that describes how many of those who received vaccine that tests positive for Covid-19 after one month. The researchers assume that The number of people who test positive for Covid-19 is binomially distributed. To continue the development of the vaccine, the researchers require that the test vaccine be included 95% security should have a positive effect against the virus. In the experiment, 10,000 people received a placebo, and 10,000 received the test vaccine. In the analysis of such an attempt should be based on the assumption that the vaccine does not work, and on that hypothesis does not hold is the basis for continuing development. (a) Find values for p and n such that X ∼ bin (n, p) if the vaccine does not work. (b) Explain why one can approximate the binomially distributed variable X with a normal distribution with expectation µ = 100 and variance σ 2 = 99. (c) Set up a hypothesis test that is relevant to the experiment. (d) Of the 10,000 people who received the test vaccine, 78 tested positive for Covid-19 after one month. Assess whether there is a basis for continuing the development of the vaccine. 4. The definition range of the function f is the convergence range of the power series: (a) Calculate the open interval where the series converges using ratios test. (b) Find an expression for the function f. (c) Write down the sequence of numbers that has the function f as its generating function. 5. In a money game, three cards are drawn f rom a deck of cards. With bet 1 you win x if there is a spade among the three cards, 2x if there are two spades and 5x whether there are three spades. (a) Explain why X = number of spades is a hypergeometrically distributed variable, and what values the parameters have. (b) If the game is to be a zero-sum game, the expected payout must be equal to the bet. Calculate what x must be for this game to be a zero-sum game. (c) An unlucky player must play five times. Let Y be the number of losses (none spar). What distribution does Y have? What is the probability that there will be a loss on exactly four of the five games? 6. A language in the alphabet {0,1} has regular expression (a) Write down all the strings in the language that have five or fewer symbols. (b) Find a final state machine that accepts the language. (c) Find a regular grammar that generates the language. 7. In a set of outcomes from an experiment, we have the three subsets A, B and C. We know about these quantities i) P (A) = 0.3 ii) P (A ∩ B) = 0.06 iii) A and B are independent iv) P (C) = 0.4 v) P (A ∩ C) = 0.12 vi) P (B ∩ C) = 0.08 vii) P (A ∪ B ∪ C) = 0.35 (a) Are A and C independent? (b) Are B and C independent? (c) Calculate P (A ∪ B). (d) Calculate P (A ∩ B ∩ C). (e) Are A and B ∪ C independent?
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1.
Let the function f : R 2 → R 2 be given by f ( x, y) = x3 − 3x+y 3 + 3 y .
(a) Find the partially derived of 1st and 2nd order.
f
f
= 3x 2 − 3,
= 3y2 + 3
x
y
(b) In which direction does the function grow most in point (2, 2) ?

 3  2 2 − 3  9 
3
The direction is f (2, 2) =  2  =   , or   (the same one).
5 
3  2 + 3 15
(c) Find and classify the stationary points of the function.
f
Since
= 3 y 2 + 3  3  0 , the function has no stationary points.
y

2. Assume that a stochastic variable X is normally distributed with unknown mean  and
2
unknown variance  . You make 10 measurements on X and get the following result:
58,73,60,64,68,60,63,67,64,77.
(a) Find a 95% confidence interval for  .
58 + 73 + 60 + 64 + 68 + 60 + 63 + 67 + 64 + 77
x=
= 65.4
10
The mean’s estimation
.
7.42 + 7.62 + 2  5.42 + 2 1.42 + 2.62 + 2.42 + 1.62 + 11.62 1622
s =
=
9
45 .
The variance’s estimation
X −
T=
S / n has the Student’s Distribution t (n − 1) , n = 10 . It follows that
The variable
2


s
s  
1622
1622 
; x + t1+ (n − 1)
; 65.4 + t0.975 (9)

 x − t1+ (n − 1)
 =  65.4 − t0.975 (9)
450
450 
n
n 

2
2
 ( 65.4 − 2.262 1.899; 65.4 + 2.262 1.899 )  ( 61.104;69.696 )
is a 95% confidence interval for  .
2
(b) Assume that you are told that X has known variance  = 36 . Find a 95% confidence
interval for µ. Explain in words why this confidence interval is less than the interval you found
above.
X −
Z=
 / n has the Standard Normal Distribution, n = 10 . It follows that
The variable



  
6
6 
; x + z1+
; 65.4 + z0.975
 x − z1+
 =  65.4 − z0.975

n
n 
10
10 

2
2
 ( 65.4 − 1.960 1.897; 65.4 + 1.960 1.897 )  ( 61.682; 69.118 )

The interval is less than the previous because knowledge of the variance  2 admits us to
evaluate the mean more definitely. Mathematically it follows from the fact what the variance of
9
1
t
(9)
7.
the standard normal distribution is less than the variance of the
-distribution,
(c) How many tests must be performed to ensure that the length of the interval should be less
than 1?

23.52

The length equals 2z1+
. It is less than on as ever n  23.522  553.2 . So, n = 554 .
n
n
2
(d) Is it possible to determine with certainty how many attempts must be perform to determine a
confidence interval of length 1 if X has unknown variance? Base the answer.
s 2 x , , xn )
We can use the empiric variance, ( 1
, to estimate the unknown variance but we do not
know nothing about the distribution of s 2 ( X1 , , X n ) . Therefore, we can’t determine n such
that 2t0.975 (n − 1)

s
would certainly be less than one.
n

3. In the development of a vaccine against Covid-19, researchers at a pharmaceutical company
try out a test vaccine on a group of people. All the people in the group tested negative for Covid19 when the experiment started. They divide the group of people involved in the experiment into
2 parts. One half (control group) receive placebo. After one month, 1% of the control group
tested positive for Covid-19. The other half of the group receives the test vaccine . Let X be the
stochastic variable that describes how many of those who received vaccine that tests positive for
Covid-19 after one month. The researchers assume that the number of people who test positive
for Covid-19 is binomially distributed. To continue the development of the vaccine, the
researchers require that the test vaccine be included 95% security should have a positive effect
against the virus. In the experiment, 10,000 people received a placebo, and 10,000 received the
test vaccine. In the analysis of such an attempt should be based on the assumption that the
vaccine does not work, and on that hypothesis does not hold is the basis for continuing
development.
(a) Find values for p and n such that X ~ Bin ( n, p ) if the vaccine does not work.
1%
= 0.01 and n = 10, 000 , the data of the control group.
100%
(b) Explain why one can approximate the binomially distributed variable X with a normal
distribution with expectation  = 100 and variance  2 = 99 .
If X ~ Bin ( n, p ) then the variance of X equals np(1 − p) = 10000  0.01 0.99 = 99 . By the de

The values are p =

Moivre–Laplace theorem the distribution Bin ( n, p ) is approximated by the normal distribution
N (  = np,  2 = np(1 − p) ) as n is large and p is separated from 0 and 1.

(c) Set up a hypothesis test that is relevant to the experiment.
The 0-hypothesis H0 : EX  100 against H1 : EX  100 .
(d) Of the 10,000 people who received the test vaccine, 78 tested positive for Covid-19 after one
month. Assess whether there is a basis for continuing the development of the vaccine.
 x − 100 
 78 − 100 
The P-value for the hypothesis H 0 equals1  
 = 
   ( −22.2 )  0 . So
/ n 
 99 / 10000 
we conclude that we have evidence against H 0 being true. It is a basis for continuing the
development of the vaccine.

1

 is the CDF of the standard normal distribution

4. The definition range of the function f is the convergence range of the power series
+

3n n
x
n =1 n
(a) Calculate the open interval where the series converges using ratiostest.
a
3n (n + 1) 1
n +1 1
The radius of convergence by the ratiotest is lim n = lim
= lim
= . Therefore,
n
+
1
n → a
n →
n3
3 n→ n
3
n +1
 1 1
the open interval where the series converges is  − ;  .
 3 3
f ( x) = 

(b) Find an expression for the function f .
The function can be expressed as an integral
3 x +
3x
+ n
+ 3 x
3
dt


3x
f ( x) =  x n =   t n −1dt =    t n −1  dt = 
= − ln(1 − t ) 0 = − ln(1 − 3 x) .
1− t
n =1 n
n =1 0

0  n =1
0
Here we use the formula for a geometric series.
(c) Write do...


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