Плотность непрерывной случайной величины равна Плотность \( \mathrm{f}(\mathrm{x})=\left\{\begin{array}{c}c \times \operatorname{arctg} x, \text { при } x \in(0,1) \\ 0, \text { при } x \nexists(0,1)\end{array}\right. \) Найти: С - ?;MX (математическое ожидание) - ?

3. Odds 8 to 5 against a person who is 40 years old living till he is 70 and 4 to 3 against another person now 50 till he will be living 80 . Probability that one of them will be alive next 30 years- \( \begin{array}{ll}\text { (A) } 59 / 9 D & \text { (B) } 44 / 91 \\ \text { (C) } 51 / 91 & \text { (D) } 32 / 91\end{array} \)

Найти вероятность попадания случайной величины \( (\mathrm{X}, \mathrm{Y}) \) в квадрат \( \frac{\pi}{6} \leq x \leq \frac{\pi}{2}, \frac{\pi}{6} \leq y \leq \frac{\pi}{2} \) Если известна функция распределения \( \mathrm{F}(\mathrm{x}, \mathrm{y})=\sin x \sin y \) \( \left(0 \leq x \leq \frac{\pi}{2} \quad, 0 \leq y \leq \frac{\pi}{2}\right) \)

The number of different permutations of all the letters of the word 'PERMUTATION' such that any two consecutive letters in the arrangement are neither both vowels nor both identical is \( \begin{array}{ll}\text { A } 63 \times 6!\times 5! & \text { B. } 57 \times 5!\times 5! \\ \text { C. } 33 \times 6!\times 5! & \text { D. } 7 \times 7!\times 5!\end{array} \)

5. An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is - (A) \( \frac{1}{3} \) (B) \( \frac{2}{7} \) (C) \( \frac{1}{21} \) (D) \( \frac{2}{23} \)

اگر احتمال رویداد A برابر با 0.3 و احتمال رویداد B برابر با 0.4 باشد، احتمال هم‌زمانی این دو رویداد را محاسبه کنید.