text: # Probability-Standard Probability Distributions (ISS Statistics Paper I (Old Subjective Pattern)): Questions 18 - 22 of 22
  
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  ## Question number: 18
  
  » Probability » Standard Probability Distributions » Poisson
  
  Appeared in Year: 2015
  
  Essay Question▾
  
  ### Describe in Detail
  
  Prove that for r = 1, 2, …, n
  
  ### Explanation
  
  We known that the L. H. S. is an incomplete gamma function and R. H. S. is a cumulative density function of Poisson distribution.
  
  The incomplete gamma function is
  
  Then
  
  provided that r is an integer. Thus recall that Γ (r) = (r… (20 more words) …
  
  ## Question number: 19
  
  » Probability » Standard Probability Distributions » Uniform
  
  Appeared in Year: 2011
  
  Essay Question▾
  
  ### Describe in Detail
  
  Let X have the continuous c. d. f. F (x). Define U = F (x). Show that both - log U arid -log (1 - U) are exponential random variables:
  
  ### Explanation
  
  Let U = F (x), then the distribution function of G of U is given by
  
  Since F is non-increasing and its continuous.
  
  G (u) =F (F -1 (u) ) implies G (u) =u
  
  Then the p. d. f is
  
  Since F is a distribution function takes value… (54 more words) …
  
  ## Question number: 20
  
  » Probability » Standard Probability Distributions » Poisson
  
  Appeared in Year: 2011
  
  Essay Question▾
  
  ### Describe in Detail
  
  Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.
  
  ### Explanation
  
  Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is
  
  So, the sum of X and Y moment generating function is
  
  because X and Y are independent
  
  ## Question number: 21
  
  » Probability » Standard Probability Distributions » Normal
  
  Appeared in Year: 2011
  
  Essay Question▾
  
  ### Describe in Detail
  
  Let the joint p. d. f. of (X, Y) be f (x, y) = e -y, 0 < x < y < ∞.
  
  Obtain the probability P (X + Y ≤ 1).
  
  ### Explanation
  
  The Joint p. d. f. is
  
  f (x, y) = e -y, 0 < x < y < ∞.
  
  Let assume X + Y =U and Y = V, then X = U-V
  
  Using Jacobian technique
  
  The range is 0 < ∞, u ≤ v < ∞
  
  The… (23 more words) …
  
  ## Question number: 22
  
  » Probability » Standard Probability Distributions » Exponential
  
  Appeared in Year: 2009
  
  Essay Question▾
  
  ### Describe in Detail
  
  Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless property.
  
  ### Explanation
  
  The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.
  
  Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is… (84 more words) …
  
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