I Year Sem- II Model Paper

        Model Paper- I
SUBJECT: STATISTICS
              Paper-II: Probability Distributions
                                        SECTION - A                                            
Write any Five questions                                                                                     5 X 4 = 20
   1. Define Bernoulli distribution. Obtain mean and variance.
   2. Derive mode of Binomial distribution.
   3. Derive m.g.f of Poisson distribution. Hence find mean and variance.
   4. State and Prove Additive property of Geometric distribution.
   5. Obtain median of Normal distribution.
   6. Define Cauchy Distribution and Write its properties.
   7. Derive m.g.f of Uniform Distribution.
   8. Explain weak law of large numbers.
                                                           SECTION - B      
Answer all 4 Questions:                                                                                         4 X 15 = 60
           
  9.(a).(i). Derive Recurrence relation for the moments of Poisson distribution.
           (ii). It is stated 2% of razor blades supplied by a company are defective.A  random sample of 200 blades is drawn from a lot. Find the probability of 3 or more are defectives.                               (OR)
      (b). Define Hyper geometric distribution. Obtain mean and variance.
10.(a). Derive m.g.f of Geometric distribution. State and prove its Lack of memory property.                                                (OR)
     (b). State and prove additive property of Negative binomial distribution. And show that Poisson distribution is a limiting case of the Negative binomial distribution.
11.(a). Calculate Even and Odd moments of Normal distribution.
                                                                        (OR)
      (b). Calculate mean and variance of Beta Distribution of 1st kind and 2nd kind.
12.(a). Obtain m.g.f and c.g.f of Exponential distribution. Also Find b1 & b2.
                                                                        (OR)
      (b). Obtain m.g.f of Gamma distribution and show that Gamma distribution tends to Normal distribution as l ® ¥.
                                                           

                                                            Model Paper II
                                                             SECTION- A
I. Answer any Five questions:                                                         5*4=20M
1.   Define Bernoulli distribution.  Obtain Mean and Variance.
2.   Define discrete uniform distribution. Obtain Mean and Variance.
3.   State and prove additive property of Binomial distribution.
4.   Define Geometric distribution. Derive its moment generating function .
5.   Define Cauchy distribution, write its properties.
6.   Write chief characteristics of normal distribution.
7.   The mean and variance of a continuous uniform random variable ‘x’ are 1.5 and 0.75 respectively. Obtain the probability density function of ‘x’.
8.   Sate weak law of large numbers. 
                                             SECTION -B
 II. Answer all questions:                                                               4*15=60M
9(a). Define negative binomial distribution. Derive its mean and variance through       expectation.                             (OR)
  (b). Define Hyper geometric distribution, give an example. Obtain its Mean          and Variance.
10(a).(i). Define Binomial distribution. Obtain moment generating function.
          (ii). Prove that Binomial distribution is the limiting case of Hyper         geometric           distribution by stating the conditions.                        
                                                                   (OR)
    (b).(i). Show that Poisson distribution satisfies the reproductive property.
          (ii).The number of monthly breakdowns of the computer is a random           variable “X” having a  Poisson Distribution with mean 2. Find the probability that this computer will function for a month
          (a).Without a breakdown  (b).With exactly one breakdown.
11(a). Define Standard normal distribution. Derive normal distribution is the        limiting case of Poisson distribution.  (OR)
     (b). (i) Define exponential distribution. Obtain its m.g.f.
      (ii) State and prove its lack of memory property.
12(a). Show that for a Normal distribution, QD :  MD  :  SD  = 10  :  12  : 15       
                                                          (OR)

(b) Define Beta distribution of 1st and 2nd kinds.  Obtain Mean and Variance of these   distributions.

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