![4.Let 𝑓 𝑥 =
2𝑥 𝑖𝑓 0 ≤ 𝑥 ≤ 2
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
be the density function of a
continuous random variable 𝑋, then
𝐸[𝑋] is_______ .](https://image.slidesharecdn.com/iimscmathematicsprobabilityandstatistics-220121171654/85/Ii-m-sc-mathematics-probability-and-statistics-8-320.jpg)
![22.Let 𝑋 be an exponential random
variable with parameter 𝜆, then
𝐸[𝑋2
] is_____.](https://image.slidesharecdn.com/iimscmathematicsprobabilityandstatistics-220121171654/85/Ii-m-sc-mathematics-probability-and-statistics-44-320.jpg)
Ii m sc mathematics probability and statistics
- 2. PROBABILITY AND STATISTICS Unit III 1.The distribution function 𝐹(𝑋) of a random variable is ______.
- 4. 2. If 𝑋 is continuous random variable whose probability density function is given by 𝐹(𝑥) = 𝐶(4𝑥 − 2𝑥 2 ), 0 < 𝑥 < 2 what is the value of 𝐶?
- 5. 3 8
- 6. 3.Let 𝑓 𝑥 = 𝑐𝑥 𝑖𝑓 0 < 𝑥 < 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 be the probability density function of a continuous random variable 𝑋, then the value of 𝑐 is______
- 7. 1 2
- 8. 4.Let 𝑓 𝑥 = 2𝑥 𝑖𝑓 0 ≤ 𝑥 ≤ 2 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 be the density function of a continuous random variable 𝑋, then 𝐸[𝑋] is_______ .
- 9. 16 3
- 10. 5.If the probability density function is given by 𝑓(𝑥) = 1, 0 < 𝑥 < 1 then the random variable is______
- 11. uniform
- 12. 6.Variance of uniform random variable is _____.
- 13. (𝛽 − 𝛼)² 2
- 14. 7.If 𝑋 is uniformly distributed over (0,5), then 𝑃{𝑋 < 2}=______ .
- 15. 2 5
- 16. 8.If 𝑋 is the normal variable then 𝑍 is defined by__________.
- 17. Z= 𝑋−𝜇 𝜎
- 18. 9.If 𝑓(𝑥) is a probability density function of a normal distribution with 𝜇 then −∞ ∞ 𝑓 𝑥 𝑑𝑥 is ____.
- 19. 1
- 20. 10.The marks secured by 400 students in a Mathematics test were normally distributed with mean 65. If 120 students got more marks above 85, the number of students securing marks between 45 and 65 is ____.
- 21. 80
- 22. 11.Expectation of negative Binomial random variable with parameters r and p is ____.
- 23. 𝑟 𝑝
- 24. 12.Variance of negative Binomial random variable is_______.
- 25. 𝑟(1 − 𝑝) 𝑝²
- 26. 13.A random variable 𝑋 is said to be geometric random variable, if the probability mass function is _____.
- 27. 𝑃 𝑋 = 𝑛 = 1 − 𝑝 𝑛−1𝑝
- 28. 14.Expected value of Geometric random variable is __________.
- 29. 1 𝑝
- 30. 15.The variance of Geometric random variable is____________.
- 31. 1−𝑝 𝑝2
- 32. 16.The expected value of the Gamma random variable is __________.
- 33. 𝛼 𝜆
- 34. 17.The variance of Gamma random variable is____________.
- 35. 𝛼 𝜆2
- 36. 18.The gamma distribution with _________ is called the chi-squared distribution with n degrees of freedom.
- 37. 𝜆 = 1 2 , 𝛼 = 𝑛 2
- 38. 19.Let 𝑋 be a gamma random variable with parameters 𝛼 and 𝜆 then 𝑉𝑎𝑟(𝑋) is_______.
- 39. 𝛼 𝜆2
- 40. 20.The mean of Beta random variable is ___________.
- 41. 𝑎 𝑎+𝑏
- 42. 21.Probability density function of exponential random variable is_____________
- 43. 𝜆 𝑒−𝜆𝑥
- 44. 22.Let 𝑋 be an exponential random variable with parameter 𝜆, then 𝐸[𝑋2 ] is_____.
- 45. 2 𝜆2
- 46. 23.Let 𝑋1, 𝑋2, . . . , 𝑋𝑛 be independent exponential random variables each having parameters 𝜆, then 𝑋1 + 𝑋2 +. . . +𝑋𝑛 is a gamma random variable with parameters__________.
- 47. (1, 𝜆)
- 48. 24.A random variable is said to have a zeta distribution if its probability mass function is given by 𝑃{𝑋 = 𝑘}=_________.
- 49. 𝐶 𝑘𝛼+1
- 50. 25.The expected number of rolls of a fair die that it takes to obtain the value 5 is______.
- 51. 0
- 52. 26.Expected value of hyper geometric random variable is__________.
- 53. 𝑛 𝑝
- 54. 27.If 𝑋 is uniformly distributed over (0,5), then 𝑃{𝑋 < 2}=______ .
- 55. 2 5