[S19-BS2] Probability and Statistics

Table of Contents

• [S19-BS2] Probability and Statistics
  • Table of Contents
  • Syllabus (lectures)
  • Grading policy
  • Assignments
    • Assignment 1: Intro
    • Assignment 2: Conditional Probability. Bayes' Law.
    • Assignment 3: Bernoulli scheme. Multinomial distribution.
    • Assignment 4: Random Variables. Discrete Distributions.
    • Assignment 5: Continuous Distributions.
    • Assignment 6: Chebyshev's Inequality and Joint Probability Distributions (Part I)
    • Assignment 7: Two-Dimensional Discrete and Continuous Distributions
    • Assignment 8: N-dimensional Distributions (Part I).
    • Assignment 9: N-dimensional Distributions (Part II).
    • Assignment 10: Characteristic Functions and Limit Theorems for Bernoulli Scheme
    • Assignment 11: Parameters Estimation. Confidence Intervals.

Syllabus (lectures)

Week	Topics
Week 1	Basic Notions of Probability. Probability Space. Conditional Probability.
Week 2	Independent Events. Inclusion – Exclusion Principle. Bayes’ Theorem.
Week 3	Bernoulli’s Trials. Polynomial Trials.
Week 4	Random Variables: Introduction to Discrete Distributions. (Tutorial: Test I)
Week 5	Different Types of Continuous Distributions and Their Properties.
Week 6	Conditional Mathematical Expectation
Week 7	Joint Probability Distributions (Part 1: $N=2$)
Week 8	Midterm (March 13)
Week 9	Joint Probability Distributions (Part 2: $N=3$)
Week 10	Joint Probability Distributions (Part 3: Advanced Tasks)
Week 11	Joint Probability Distributions (Part 4: Normal Distribution)
Week 12	Characteristic Functions. Limit Theorems. (Tutorial: Test II)
Week 13	Maximum Likelihood Estimation. Parameter Estimation.
Week 14	Rao-Blackwell Theorem. Confidence Intervals.

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Grading policy

Contents

Item	Points	Comment
TA gratitude	5	depending on TA
Tests (90 min.)	10x2	week 4 and week 12
Mid-term exam	25	week 8
Final exam	45+5	45 – written, 5 – non-obligatory 'spoken' after written (week 16)

Final grade scale

Grade	Interval
A	80+
B	60-79
C	40-59

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Assignments

Stepik page of this course

Assignment 1: Intro

(Up to Jan. 30)

(Tutorial: 23, 32, 46, 59, 81, 89, 90, 102)

Set	Page	Tasks	Topic
2.2	42	10, 15, 18	Events
2.2	51	26, 31, 37, 44, 45	Counting Sample Points
2.5	59	52, 63, 72	Additive Rules
2.6	69	73, 91, 93, 94	Conditional Probability and Product Rule
2.7	76	99, 101	Bayes' Rule
~	77	104, 109, 113, 125	Review Excercises

See also:
Base rate fallacy [en/ru]

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Assignment 2: Conditional Probability. Bayes' Law.

(Up to Feb. 6)

1. Prove that
   (a)

   $P(AB)\ge P(A)+P(B)-1$;
   (b)
   $P(A_1{A}_2\dots A_n)\ge P(A_1)+\dots +P(A_n)-(n-1)$.

2. Let

   $P(A)=0$. Prove that
   $P(A+B)=P(B)$. Is the opposite statement also true?

3. Let

   $P(A)=1$. Prove that
   $P(AB)=P(B)$. Is the opposite statement also true?

4. Prove that if

   $P(A)=\frac{2}{3}$,
   $P(B)=\frac{3}{4}$, then
   $P(A|B)\ge \frac{5}{9}$.

5. The letters of a word ``ITERATION'' are arranged in a random order. Find the probability that the same word has been obtained this way.

6. Let

   $A$ and
   $B$ be independent events. Prove that (a)
   $A$ and
   $\bar{B}$ are independent; (b)
   $\bar{A}$ and
   $\bar{B}$ are independent.

7. Two players take turns flipping a coin. A player who gets heads first wins the game. What is the probability for the second player to win the game?

8. A fair die is rolled five times. What is the probability to get the same number at least three times in a row?

9. (a) A coin is flipped until the sequence ``tails, heads, tails'' is observed. What is the probability that the coin has been flipped exactly 7 times?

   (b) A coin is flipped until the sequence ``tails, tails, tails'' is observed. What is the probability that the coin has been flipped exactly 7 times?

10. Four black balls and eight white balls are arranged into a line. Find the probability that
    (a) the first white ball is situated in the third place;
    (b) the last white ball is situated in the ninth place;
    (с) the first black ball is situated in the third place and the last black ball is situated in the ninth place.

11. There are 10 white and 20 black balls in the first urn and 10 white balls and 10 black balls in the second urn. Four balls from the first urn and six balls from the second urn are selected at random and placed into the third urn (that was initially empty). Determine the probability that a ball randomly drawn from the third urn happens to be white.

12. There are 12 white balls and 8 black balls in the first urn; 8 white balls and 4 black balls in the second urn; 10 white balls in the third urn. 6 balls from the first urn, 5 balls from the second one, and 4 balls from the third one, all of them chosen at random, are transferred into the fourth urn that was empty at first. After that 2 arbitrary balls are simultaneously drawn from the fourth urn. Both of them have turned out to be white. Find the probability that these balls originate from different urns.

13. A message is transmitted via some communication channel. It can contain one of the three letter sequences:

    $AAAA$,
    $BBBB$,
    $CCCC$. On average, the last sequence is transmitted two times less often than the second one and three times less often than the first one. Due to interference a sequence can be distorted. The probability for a letter to be transmitted correctly is
    $0.8$, whereas the probability that one letter turns into another one is
    $0.1$. (Thus if
    $A$ is transmitted there is a
    $0.8$ probability to receive
    $A$,
    $0.1$ probability to receive
    $B$ and
    $0.1$ probability to receive
    $C$). The distortions for each of the positions in the sequence do not depend on one another. Determine the probabilities that
    (a)
    $AAAA$ was transmitted, if
    $CACB$ has been received.
    (b)
    $CCCC$ was transmitted, if
    $CACB$ has been received.

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Assignment 3: Bernoulli scheme. Multinomial distribution.

(Up to Feb. 13)

1. A coin is flipped 10 times. What is the probability that there are exactly 7 heads and 3 tails?

2. Every projectile hits the target with probability 0.4. If the target was hit once, the probability to be destroyed is 0.3, twice – 0.7. If the target was hit 3 times or more it will be destroyed for sure. Find the probability that the target was destroyed if

3. On the average 2% of devices are broken. What is the probability that not less than 2 devices out of 6 are broken?

4. There are 10 devices. Every device independently breaks down with probability 0.15. Find the probability that:
   (a) exactly 3 devices broke down
   (b) exactly 3 devices broke down if it is known that at least 1 device broke down

5. A device is broken if at least five 1st-type-sensors or at least two 2nd-type-sensors are broken. From the set of all broken sensors there are 70% of type "1" and 30% of type "2" are broken (on the average). It is known that 5 sensors are broken.
   (a) What combination of broken sensors is most likely?
   (b) What is the probability that a device is broken?

6. The probability to hit an aim is

   $p$. Find the probability that the second out of five shoots was failed if:
   (a) the aim was hit exactly 3 times
   (b) at least one shoot was failed

7. Nine persons entered a nine-floor house with an elevator. What is the probability that the elevator will stop on each floor. Every person can exit on each floor with the same probability.

8. Find the probability that in Bernoulli trials with

   $p=0.4$ three successes will happen earlier than five fails.

9. The first player rolls 6 dices and wins if "1" occurs at least once.
   The second player rolls 12 dices and wins if "1" occurs at least twice.
   Who has more chances to win?

10. A bar of length L is divided into three random parts. Determine the probability that a triangle may be formed from these parts.

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Assignment 4: Random Variables. Discrete Distributions.

(Up to Feb. 20)

1. Let

   $\xi \sim \left(\begin{array}{ccccc}-2& -1& 0& 1& 2\\ 0.1& 0.3& 0.2& 0.2& 0.2\end{array}\right)$
   Plot the cumulative distribution function of
   $\xi$.
   a) Find
   $M\xi$
   b) Find
   $D\xi$
   c) Find
   $\sigma$

2. A coin is flipped

   $\xi$ times until two "tails" or two "heads" is observed. 
   Plot the cumulative distribution function of
   $\xi$.
   a) Find
   $M\xi$
   b) Find
   $D\xi$

3. A random variable

   $\xi$ has three possible values:
   $-1,0,1$. It is known that
   $M\xi =0$ and
   $D\xi =0.5$. Find
   $P(\xi =-1)$, 
   $P(\xi =0)$ and 
   $P(\xi =1)$.

4. Let

   $\xi \sim \left(\begin{array}{cccc}0& 1& 2& 3\\ ?& 1/3& ?& 1/6\end{array}\right)$
   a) Find
   $D\xi$ if it is known that
   $M\xi =1$.
   b) Find
   $M\xi$ if it is known that
   $D\xi =11/12$.

5. From a deck of cards (52 cards) randomly (without replacement) a card is taken until "ace of diamonds" is observed.
   a) How many cards will be taken on average?
   b) What is the probability that not more than 26 (a half) card will be taken?

6. Find a formula for

   $D\xi$ for the uniform distribution.

7. Three students independently pass an exam with the probabilities

   $0.9$,
   $0.8$ and
   $0.7$. Let
   $\xi$ is a total number of students passed the exam.
   Find
   $M\xi$ and
   $D\xi$.

8. Ten people entered to an elevator of 9-floor house. How many times the elevator will stop on average?

9. A number from all three-digit numbers is picked randomly. Let

   $\xi$ is a number of different digits in our number. Find
   $M\xi$.

10. Find parameters

    $(n,p)$ of Binomial Distribution for a random variable
    $\xi$ or claim that
    $\xi$ is not from Binomial Distribution if:
    a)
    $M\xi =6$ and
    $D\xi =3$
    b) 
    $M\xi =7$ and
    $D\xi =4$

11. For which

    $p$ a random variable
    $\xi \sim Bin(n,p)$ has maximum dispersion?

12. A coin is flipped several times until 3 "tails" (so the first player wins) or 3 "heads) (so the second player wins) is observed (in total). Let

    $\xi$ is a number of times that coin was flipped. Find
    $M\xi$.

13. Let

    $\xi \sim G(p)$ and
    $P(\xi =2)=2/9$. Find
    $P(\xi =6)$.

14. The probability to hit an aim is

    $p=0.7$. Three misses are allowed.
    a) Find the average number of shoots [before three misses].
    b) Find the probability to shoot exactly eight times.

15. A fair dice is rolled until every number appears at least once. How many times (on the average) a fair dice will be rolled?

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Assignment 5: Continuous Distributions.

(Up to Feb. 27)

1. Inside a sphere of radius

   $r$ a point
   $M$ is picked randomly. Let
   $\xi$ is a distance from M to the sphere (to its surface).
   Find
   $F(x)$,
   $f(x)$,
   $M\xi$ and
   $D\xi$.

2. Let

   $f(x)=\{\begin{array}{l}C(1-|x|),\text{if }|x|<1,\\ 0,\text{if }|x|>1;\end{array}$
   a) Find
   $C$ so that
   $f(x)$ is PDF for some distribution.
   b) Find
   $M\xi$ using found
   $C$.
   c) Find
   $D\xi$ using found
   $C$.

3. Let

   $f(x)=C{e}^{-|x|}$ for
   $x\in \mathbb{R}$.
   a) Find
   $C$ so that
   $f(x)$ is PDF for some distribution.
   b) Find
   $M\xi$ using found
   $C$.
   C) Find
   $D\xi$ using found
   $C$.

4. Let

   $\xi \sim U(0,4)$. Find:
   a)
   $P(\xi <M\xi )$
   b)
   $P(\xi >\sqrt{D\xi })$
   c)
   $P(-5\le \xi \le 5)$

5. Let

   $\xi \sim U(a,b)$ (
   $a$ and
   $b$ are not given). It is known that
   $P(\xi <1)=\frac{1}{2}$ and 
   $P(\xi <2)=\frac{2}{3}$. Find
   $F(x)$,
   $f(x)$,
   $M\xi$ and
   $D\xi$.

6. Let

   $\xi \sim U(-1,5)$. Find
   $M[(\xi -1)(3-\xi )]$.

7. Let

   $\xi \sim E(\lambda )$.
   Which event is more likely to happen: {
   $\xi <M\xi$} or {
   $\xi >M\xi$}?

8. Let

   $\xi \sim E(\lambda )$ (
   $\lambda$ is not given). It is known that
   $P(|\xi -M\xi |<1)=6/7$.
   Find
   $M\xi$.

9. Let

   $\xi \sim \mathcal{N}(1,4)$. Find:
   a)
   $P(-3<\xi <1)$;
   b)
   $P(-\mathrm{\infty }<\xi <-2)$;
   c)
   $P(3<\xi <\mathrm{\infty })$.

10. Let

    $\xi \sim \mathcal{N}(\mu ,{\sigma }^2)$. It is known that 
    $P(1<\xi <7)=P(7<\xi <13)=0.18$. Find
    $M\xi$ and
    $D\xi$.

11. Let

    $\xi \sim \mathcal{N}(-2,9)$. Find
    $M[(3-\xi )(5+\xi )]$.

12. Prove that if

    $\xi \sim U(a,b)$ then for
    $\alpha >0$
    
    $\eta =\alpha \xi +\beta \sim U(\alpha a+\beta ,\alpha b+\beta )$.

13. Prove that if

    $\xi \sim U(-a,a)$ then
    
    $\eta =|\xi |\sim U(0,a)$.

14. Let

    $\xi \sim \mathcal{N}(\mu ,{\sigma }^2)$ and it is known that
    $P(1<\xi <7)=P(3<\xi <9)$. What is maximum possible value of
    $P(1<\xi <3)$?

15. Let

    $\ln \xi =\eta \sim \mathcal{N}(0,1)$. Prove that
    $M({\xi }^n)=M(e^{n\eta })=e^{n^2/2}$.
    Note: Log-normal distribution

16. Let

    $\ln \xi =\eta \sim \mathcal{N}(0,1)$. So
    $\xi =e^{\eta }$. Prove that PDF for
    $\xi$ is
    $f(x)=\frac{1}{\sqrt{2\pi }}\frac{1}{x}{e}^{-\frac{1}{2}(\ln x{)}^2}$.

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Assignment 6: Chebyshev's Inequality and Joint Probability Distributions (Part I)

(Up to Mar. 6)

1. Average water consumption in a city is 50000 litres per day. Estimate the probability that real water consumption is less than tripple average.

2. Six fair dices are rolled. Let

   $\xi$ is the least digit occured on these six dices. Find
   $E\xi$.

3. $N$ fair dices are rolled. Let
   $\xi$ is the least digit occured on these
   $N$ dices. Find
   $E\xi$ if
   $N\to \mathrm{\infty }$.

4. Ten people entered to an elevator of 9-floor house. Let

   $\xi$ is a floor number on which the first person will exit. Find
   $E\xi$.

5. Ten people entered to an elevator of 9-floor house. Let

   $\xi$ is a floor number on which the last person will exit. Find
   $E\xi$.

6. A shooter has 25 bullets. The probability to hit a target is

   $p=0.96$. The shooter shoots until the first miss or he is out of bullets.
   Let
   $\xi$ a number of spent bullets.
   Find
   $E\xi$.

7. Find

   $E[(\xi +\eta {)}^2|\xi =1]$ if:

   $\xi \eta$	$-1$	$1$
   $0$	$1/6$	$1/6$
   $1$	$1/6$	$1/2$

8. In a city there is sunny with the probability

   $p=0.6$ and it is cloudy with the probability
   $q=0.4$. How many days on the average a person should be in the city to see both sunny and cloudy days?

9. A coin is flipped until "tails, tails" combination is observed.
   How many times on average a coin will be flipped?

10. A fair dice is flipped until "66" is observed.
    How many times on average a dice will be flipped

11. A fair dice is flipped until "666" is observed.
    How many times on average a dice will be flipped?

12. A fair dice is flipped until "6" is observed.
    Let

    $S$ is a sum of scores. Find
    $ES$.

13. Twelve people entered to an elevator of 12-floor house. How much people on average will be in the elevator after a certain person X exit the elevator.

14. Let

    $\xi \sim \left(\begin{array}{ccc}-a& 0& a\\ p& 1-2p& p\end{array}\right)$, where
    $a>0$ and
    $0<p<0.5$.
    Prove that:
    
    $P(|\xi -E\xi |\ge a)=Var(\xi )/a^2.$

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Assignment 7: Two-Dimensional Discrete and Continuous Distributions

(Up to Mar. 13)

1. $N$ fair dices are rolled.
   Let
   $X$ is a quantity of '6' appeared.
   Let
   $Y$ is a quantity of even numbers appeared.
   Find
   ${\rho }_{X,Y}$ (Correlation coefficient).

2. Let

   $\xi$ and
   $\eta$ are random variables with the joint distribution below.
   Find
   $Var(\eta |\xi =1)+Var(\xi |\eta =0)$.
   Also find the law of distribution of
   $\xi +\eta$,
   $\xi \eta$ and
   $(\xi \eta ,\xi +\eta {)}^T$.

   $\xi$ \ $\eta$	$-1$	$0$	$1$
   $-1$	$1/8$	$1/12$	$7/24$
   $1$	$1/3$	$1/6$	$0$

3. Ten people entered to an elevator of 9-floor house.
   Let

   $X$ is a number of people exited at eighth floor.
   Let
   $Y$ is a number of people exited at ninth floor.
   Find
   ${\rho }_{X,Y}$ (Correlation coefficient).

4. Let

   $Z$ a random variable from uniform distribution with possible values
   $-1,0,1$.
   Let
   $X=1-Z^{1000}$ and
   $Y=1-Z^{1001}$. 
   Are
   $X$ and
   $Y$ independent? Correlated?

5. Let

   $a$,
   $b$,
   $c$ and
   $d$ are some constants.
   Prove that cov
   $(aX+b,cY+d)$ =
   $ac$ cov
   $(X,Y)$.

6. A random vector

   $(\xi ,\eta {)}^T$ has uniform distribution in
   
   $D=${
   $x^2+y^2\le 25,x\ge 0$}.
   a) Find
   $E(\xi |\eta =4)$;
   b) Find
   $Var(\xi |\eta =4)$.

7. Let

   $\xi \sim N(0,1)$ and
   $\eta ={\xi }^2$.
   Are
   $\xi$ and
   $\eta$ dependent? Correlated?

8. Let

   $(\xi ,\eta {)}^T$ a random vector with covariance matrix:
   
   $K=\left(\begin{array}{cc}2& -3\\ -3& 6\end{array}\right)$
   a) Find
   $Var(-\xi -2\eta )$;
   b) Find
   $Var(3\xi -2\eta +2)$.

9. Let

   $\xi \sim N(0,1)$ and 
   $\eta \sim N(0,1)$ are independent.
   Prove that
   ${\xi }^2+{\eta }^2\sim E(1/2)$.

10. Let

    $X_1\sim E(\lambda )$ and
    $X_2\sim E(\lambda )$.
    Let
    $Y=X_1/X_2$.
    Find PDF
    $f_Y(x)$.

11. Let

    $X_1\sim N(0,1)$ and
    $X_2\sim N(0,1)$.
    Let
    $Y=X_1/X_2$.
    Find PDF
    $f_Y(x)$.

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Assignment 8: N-dimensional Distributions (Part I).

1. Let

   $X$ and
   $Y$ be independent exponentially distributed random variables with parameters
   $\lambda$ and
   $\mu$ respectively. Let us consider random variables
   $U=min\{X;Y\}$,
   $V=max\{X;Y\}$. Determine whether
   $U$ and
   $V$ are independent. Is random variable
   $U$ independent from the event
   $X>Y$ (i.e. from the indicator random variable of this event:
   $I_{\{X>Y\}}$)?

2. A marksman is shooting at a round target (centered at the origin). The vertical and horizontal coordinates of a point

   $M$ where the bullet hits the target are independent
   $N(0;1)$ random variables. Determine the probability density of a random variable equal to the distance from point
   $M$ to the origin.

3. Let

   $X_1$ and
   $X_2$ be independent random variables with Cauchy distribution
   $(f_X(t)=\frac{d}{\pi (d^2+x^2)})$.
   a) Prove that
   $EX$ does not exist.
   b) Prove that
   $\frac{X_1+X_2}{2}$ is also a Cauchy distributed random variable.

4. Let

   $X_1,X_2,\dots ,X_n$ be independent exponentially distributed random variables, the parameter of the distribution
   $\lambda$ being the same for all of them. Find the distribution for
   a)
   $Y_n=max[X_1,X_2,\dots ,X_n]$;
   b)
   $Z_n=\sum _{j=1}^n\frac{X_j}{j}$.

5. Let

   $X_k$ be a sequence of independent random variables uniformly distributed on
   $[0;1]$, and
   $U_n=\underset{1⩽j⩽n}{max}{X}_j$,
   $V_n=\underset{1⩽j⩽n}{min}{X}_j$.
   a) Find joint probability density of
   $(U_n;V_n)$.
   b) Find the correlation coefficient of
   $U_n$ and
   $V_n$ and show that it approaches 0 as
   $n\to \mathrm{\infty }$.

6. The probability density of a random vector

   $(X;Y)$ is given by
   $$f_{X;Y}(x;y)=\frac{1}{2\pi \sqrt{1-r^2}}\exp (-\frac{1}{2(1-r^2)}(x^2-2rxy+y^2)),$$
   where
   $r\in (-1;1)$ is some number.
   a) Find marginal distributions of
   $X$ and
   $Y$.
   b) Find the correlation between
   $X$ and
   $Y$.

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Assignment 9: N-dimensional Distributions (Part II).

1. Let

   $\xi$ =
   $(X,Y,Z{)}^T$ ~
   $N(\mu ,K)$.
   $\mu =\left(\begin{array}{c}1\\ 0\\ -2\end{array}\right)$,
   $K=\left(\begin{array}{ccc}2& 3& -1\\ 3& 6& -1\\ -1& -1& 1\end{array}\right)$.

   a) Find

   $H(\xi )$ (hint);
   b) Find
   $f_1(x),f_2(y),f_3(z),f_{13}(x,z)$;
   c) Find
   $P(2X-3Y-Z<9)$;
   d) Find
   $P(|2Y-5Z|<16)$.

2. Let

   $\xi$ =
   $(X,Y{)}^T$ ~
   $N(\mu ,K)$.
   $\mu =\left(\begin{array}{c}-3\\ 1\end{array}\right)$,
   $K=\left(\begin{array}{cc}1& -1\\ -1& 4\end{array}\right)$.

   a) Find

   $P(Y<3|X=0)$;
   b) Find
   $P(|Y+4|<2|X=2)$.

3. Let

   $X_1,X_2,...,X_{100}$ ~
   $N(0,1)$ – are independent.
   Let
   $Y_1=X_1+X_2+...+X_{40}$ and 
   $Y_2=X_1+X_2+...+X_{100}$.
   Find PDF of
   $Y=(Y_1,Y_2{)}^T$.

4. Let

   $X$ ~
   $N(0,1)$ and 
   $Y$ ~
   $N(0,1)$ are independent. Find
   $P(X<3Y)$.

5. Let

   $X$ ~
   $N(\mu ,\sigma )$ and 
   $Y$ ~
   $N(\mu ,\sigma )$.
   Prove that
   $\xi =X+Y$ and
   $\eta =X-Y$ are independent.

6. Let

   $X$ ~
   $N(0,1)$ and 
   $Y$ ~
   $N(0,1)$ with correlation coefficient equals to
   $\rho$.
   Find
   $E(X^3{Y}^3)$ as a function of
   $\rho$.

7. Let

   $X_1,X_2,...,X_n$ ~
   $N({\mu }_1,{\sigma }_1)$ and 
   
   $Y_1,Y_2,...,Y_m$ ~
   $N({\mu }_2,{\sigma }_2)$. 
   {
   $X_1,X_2,...,X_n$} and {
   $Y_1,Y_2,...,Y_m$} are independent samples.
   Find PDF of:
   a)
   $S=\frac{1}{{\sigma }_1^2}\sum _{i=1}^n{X}_i+\frac{1}{{\sigma }_2^2}\sum _{j=1}^m{Y}_j$;
   b)
   $D=\frac{1}{n}\sum _{i=1}^n{X}_i-\frac{1}{m}\sum _{j=1}^m{Y}_j$.

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Assignment 10: Characteristic Functions and Limit Theorems for Bernoulli Scheme

1. Let

   $X$
   $\sim$
   $N(0,1)$. Find
   $E[\cos X]$.

2. Let

   $\phi (t)=\frac{\cos (t)}{1+t^2}$ is characteristic function of
   $X$.
   Find
   $E[X]$ and
   $Var(X)$.

3. Prove that a function

   $\phi (t)=e^{-t^4}$ is not a characteristic function.

4. Let

   $X\sim E(\lambda )$ and
   $Y\sim E(\lambda )$ are independent.
   Find PDF of
   $Z=X-Y$ using characteristic functions.

5. Let

   $X_1,X_2,...,X_n$ are all distributed as
   $N(\mu ,{\sigma }^2)$.
   Let
   $\bar{X}=\frac{1}{n}(X_1+X_2+...+X_n)$.
   Find
   $E[\bar{X}]$ and
   $Var(\bar{X})$.

Poisson limit theorem
De Moivre–Laplace theorem
De Moivre–Laplace theorem (integral case)

6. A book has 500 pages and 400 misprints. Let all misprints can occur independently on any page with equal probabilities. Estimate the probability that on the seventeenth page there will be more than one missprint.

7. The probability that there are no raisins in a bun is

   $0.003$. Find the probability that in 1000 buns there are:
   a) no buns without raisins
   b) 3 buns without raisins
   c) more than 2 buns without raisins

8. Every citizen independently votes for the candidate A with the probability equals

   $0.7$ and for the candidate B with the probability equals
   $0.3$. There are 5000 citizens.
   a) Estimate the probability that candidate A will have 1900 more votes than candidate B;
   b) Estimate the probability that candidate A will have not less than 1900 more votes than candidate B.
   (hint)

9. What is the minimum number of times a coin must be flipped to have 95% confidence that deviation of relative frequency of "tails" from the probability that "tails" occurs is not more than

   $0.01$.

10. A fair dice is flipped 1000 times. With the probability

    $p$ we will have from
    $a$ to
    $b$ "six"-scores. Find the smallest [a,b] if:
    a)
    $p=0.95$;
    b)
    $p=0.99$.
    Note: such interval [a, b] is also called Confidence interval with the level of confidence equals to 95% (99%).

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Assignment 11: Parameters Estimation. Confidence Intervals.

1. Let {

   $x_n$} = {
   $1.3,4.2,3.7,0.6,0.3,1.5,2.4$} –
   $n$ independent draws from a distribution. Find maximum likelihood estimation:
   a) Poisson distribution. Parameter:
   $\lambda$;
   b) Exponential distribution. Parameter:
   $\lambda$;
   c) Normal distribution. Parameter:
   $\mu$;
   d) Normal distribution. Parameter:
   $\sigma$.

2. Let {

   $x_n$} = {
   $1.3,4.2,3.7,0.6,0.3,1.5,2.4$} –
   $n$ independent draws from a distribution. Find sufficient statistics:
   a) Uniform distribution
   $[0,\theta ]$. Parameter:
   $\theta$;
   b) Poisson distribution. Parameter:
   $\lambda$;
   c) Normal distribution. Parameter:
   $\mu$;
   d) Normal distribution. Parameter:
   $\sigma$.

3. Explain what is sufficient statistics. What is factorization criterion?
   Let

   $X_1,...,X_n$ are independent random variables from the same distribution with PDF:
   $f(x,\theta )=\theta e^{-\theta x+1},x\ge 1/\theta .$
   Using factorization criterion find sufficient two-dimensional statistics
   $T$ for parameter
   $\theta$.

4. Let

   $(x_1,...,x_6)$ – is a sample from
   $U(0,\theta )$ (
   $\theta \in [1,2]$ is not given).
   Find bias of
   $\theta$ with variance less than
   $0.1$.

5. Let

   $(x_1,x_2,...,x_{100})$ – is a sample from
   $N(\mu ,\sigma )$.
   Find 90%-confidence interval
   $[\mu -\mathrm{\Delta },\mu +\mathrm{\Delta }]$ for
   $\mu$
   a) if sample mean
   ${\bar{x}}_{100}$ is equal to
   $1$ and
   $\sigma =1$;
   b) if sample mean
   ${\bar{x}}_{100}$ is equal to
   $1$ and adjusted sample variance is equal 4.

Useful resource

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tags: Innopolis University