This document provides definitions and key properties for 27 different probability distributions and statistical concepts. Some of the distributions and concepts covered include: - The distribution function of a random variable is a non-decreasing function. - Parameters and formulas are provided for the normal, negative binomial, geometric, gamma, beta, exponential, and hypergeometric distributions. - Expected values and variances are given for several continuous and discrete distributions. - The relationship between the normal distribution and the z-score is defined.

2. PROBABILITY AND STATISTICS
Unit III
1.The distribution function 𝐹(𝑋) of a
random variable is ______.

3. a non-decreasing function

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