# 1. Problem 21

## 1.1. Question

We toss 12 coins. Let X denote number of Heads. Find $$F_X(\frac{41}{10})$$.

1. $$\frac{397}{2048}$$
2. $$\frac{1}{4097}$$
3. $$\frac{1585}{4096}$$
4. $$\frac{397}{4096}$$
5. $$\frac{1}{683}$$

1 is true; the rest is false

Binomial distribution

$$(\frac{1}{2})^{12} (\binom{12}{0}+\binom{12}{1}+\binom{12}{2}+\binom{12}{3}+\binom{12}{4}) = \frac{397}{2048}$$

# 2. Problem 22

## 2.1. Question

Distribution of the characteristic $$X$$ has density:

$$f_p(x) = \begin{cases} (p+1)x^p & 0 \leq x \leq 1 \\ 0 & \text{for the remaining x} \end{cases}$$

Let's consider simple random variable $$(X_1,\cdots,X_n)$$. Let $$\overline{X_n^2} = \frac{1}{n} \sum_{i=1}^n X_i^2$$, $$\overline{X_n^3} = \frac{1}{n} \sum_{i=1}^n X_i^3$$. The statistic shown below is maximum likelihood estimator of the parameter p:

1. $$\frac{3}{1-\overline{X_n^3}} -4$$
2. $$\frac{2}{1-\overline{}X_n^2} -3$$
3. $$- \frac{n}{\sum_{i=1}^n ln(X_i)} -1$$
4. $$- \frac{n}{\sum_{i=1}^n ln(X_i)} +1$$
5. $$\frac{n}{\sum_{i=1}^n ln(X_i)} -1$$

3 is correct; the rest is false

$$L(x_1,\cdots,x_n;p) = \prod_{i=1}^n (p+1)x_i^p = (p+1)^n \times \prod_{i=1}^n x_i^p$$

$$\ln (L(x_1,\cdots,x_n;p)) = n \ln (p+1) + p \sum_{i=1}^n x_i$$

We take derivative, and look for place, when it is equal to zero:

$$0 = \frac{n}{p+1} + \sum_{i=1}^n \ln (x_i)$$

Therefore:

$$p = - \frac{n}{\sum_{i=1}^n \ln (x_i)} - 1$$

# 3. Problem 23

## 3.1. Question

Random variable $$X$$ has uniform distribution on the interval $$(0,3)$$, and $$Y$$ is independent and has two-point distribution $$P(Y=-1) = 1-P(Y=1)=\frac{2}{3}$$. $$p = P(X+Y<\frac{3}{2})$$.

1. $$p < \frac{1}{3}$$
2. $$p \geq \frac{3}{4}$$
3. $$p \geq \frac{1}{3}$$
4. $$p > P(X+Y<\frac{17}{10}$$

3 is true; the rest is false; 2 is impossible

The actual probability: $$p = \frac{11}{18}$$

# 4. Problem 24

## 4.1. Question

Random variable $$X$$ has uniform distribution on $$[-1,1]$$. Random variable $$(X,3X-4)$$ has distribution:

1. singular
2. mixed
3. of continuous type
4. of discrete type

3 is true; the rest is false

# 5. Problem 25

## 5.1. Question

Geiger-Müller counter and source of radiation are placed so, that the probability of registering by this counter an emitted particle is equal to 0.0004. During observation period radioactive specimen radiated 5000 particles. Probability $$p$$of the event that counter registered at most two particles, is:

1. $$p = 0.67668$$
2. $$p \leq 0.67668$$
3. $$p = 0.27067$$
4. $$p \geq 0.94735$$
5. $$p = 0.54134$$

2 is true; the rest is false; 1 is impossible (is it?); 1 and 2 are suspicious

More precisely $$p = 0.6766764162$$.

**Poisson distribution**

# 6. Problem 26

## 6.1. Question

Let random variables $$X_i, i \geq 1$$ be independent, and let them have the following identical distributions:

$$P(X=x) = \begin{cases} \frac{1}{6} & \text{for  x=-1 \vee x=5 } \\ \frac{1}{3} & \text{for x=1 \vee x=-5} \end{cases}$$

$$cos(\pi \frac{\text{number 5 and -5 in n observations}}{n})$$ converges to:

1. $$- \frac{1}{2}$$
2. $$1$$
3. $$0$$
4. $$\frac{1}{2}$$

3 is correct; the rest is false

**Law of large numbers**

# 7. Problem 27

## 7.1. Question

Characteristic $$X$$ has density

$$f_p = \begin{cases} p(1-|x|)^{p-1}/2 & \text{for  |x| \leq 1 } \\ 0 & \text{for |x| > 1} \end{cases}$$

$$p>1$$

Let $$(X_i,\cdots,X_n)$$ be simple random sample. Let's denote: $$\overline{g(X)}_n = \frac{1}{n} \sum_{i=1}^n g(X_i)$$, $$p \int_0^1 x^n(1-x)^{p-1} = I(n,p)$$. It is known that $$I(2,p) = \frac{2}{(p+2)(p+1)}$$, $$I(3,p) = \frac{6}{(p+3)(p+2)(p+1)}$$, $$p \int_o^1 (1-x)^{p-1} ln(1-x)dx = - \frac{1}{p}$$. Which statement is true?

1. $$\frac{1}{\overline{|X|}_n}$$ is a consistent estimator of parameter p
2. $$p \overline{ln |1-|X||}_n \rightarrow 1$$ with probability $$1$$
3. $$(1+p)\overline{X^2}_n \rightarrow \frac{1}{p+2}$$ with probability $$1$$
4. $$cos(\pi (p+1) \overline{|X|}_n) \rightarrow 1$$ with probability $$1$$
5. $$(p+3)(p+2) \overline{X^3}_n \rightarrow 0$$ with probability $$1$$

1. #red|NO ANSWER## It looks like there is no correct answer.

# 8. Problem 28

## 8.1. Question

Distribution of the characteristic X has density:

$$f_p(X) = \begin{cases} (p+1)x^p & 0 \leq x \leq 1 \\ 0 & \text{for the remaining x} \end{cases}$$

Let's onsider simple random sample $$(X_1, \cdots, X_n)$$. Let $$\overline{X_n^2} = \frac{1}{n} \sum_{i=1}^n X_i^2$$, $$\overline{X_n^3} = \frac{1}{n} \sum_{i=1}^n X_i^3$$. Are the statistics shown below consistent estimators of the prameter p?

1. $$\frac{3}{1-\overline{X_n^3}} - 3$$
2. $$\frac{3}{1-\overline{X_n^3}} - 4$$
3. $$\frac{2}{1-\overline{X_n^3}} - 4$$
4. $$\frac{2}{1-\overline{X_n^2}} - 2$$
5. $$\frac{\overline{X_n}+\overline{X_n^3}}{2}$$

2 is true; the rest is false

$$\overline{X_n^3} \rightarrow \frac{p+1}{p+4}$$

$$\frac{3}{1-\overline{X_n^3}} - 4 \rightarrow \frac{3}{1-\frac{p+1}{p+4}} - 4 = \frac{3}{\frac{3}{p+4}} - 4 = p$$

# 9. Problem 29

## 9.1. Question

Random variables $$(X,Y)$$ have uniform distribution on set $$D=\{ (x,y) : -3 \leq x \leq 0; 0 \leq y \leq x+3 \} \cup \{ (x,y) : 0 \leq x \leq 3; x \leq y \leq 3 \}$$. The best mean squares approximation of Y by a linear function of X is:

1. $$\frac{1}{2}X + \frac{3}{2}$$
2. $$\frac{3}{2}$$
3. $$- \frac{1}{2} X + \frac{3}{2}$$
4. $$\frac{1}{4}X - \frac{3}{8}$$

1 is true; the rest is false

$$E(X) = 0$$; $$E(Y) = \frac{3}{2}$$; $$E(X^2) = \frac{3}{2}$$; $$E(XY) = \frac{3}{4}$$

Can be answered by looking at the plot.

# 10. Problem 30

## 10.1. Question

Random variables $$X$$ and $$Y$$ are independent with identical uniform distribution on the interval $$(0,3)$$. Let $$Z=X+Y$$, $$T = X-Y$$. The error $$r$$ of approximation of $$Z$$ by linear regression of $$Z$$ on $$T$$ is:

1. $$r = 0.9$$
2. $$r = \frac{3}{2}$$
3. $$r > \frac{9}{5}$$
4. $$r = \frac{9}{8}$$
5. $$r \in (0;0.9)$$

$$r = Var(T) (1 - \rho_{Z,T}^2) = Var(T) = E(X^2)-2 E(XY)+E(Y^2) = \frac{3}{2}$$