BSc Sem IV 2022 2023 Mar 2023 STATISTICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

• (2) Figures to the right indicate full marks

1. Q1 Attempt all sub questions: 20 marks
   • (A) State True or False and correct if necessary: 10
   • (i) Ifa random variable follows Uniform distribution over (-1, 1), then its mean is 0
   • (ii) Ifa r.v. follows Exponential distribution with parameter 3, its mean is 1/3
   • (iii) If then 5" central moment is equal to 5
   • (iv) Sqaure of Standard Normal Variable has Chi-Square distribution with n degrees of
   • (v) If F(x) = 1 — then the distribution is symmetric
   • (B) Answer the following. 10
   • (i) Ifa r.v. follows Uniform distribution over (-5, 5), write down its probability density
   • (ii) Identify the distribution and find its mean if p.d.f. is given by (ili) X~N(18, 25), what is the third quartile of the distribution,
   • (iv) When do you apply Correction? Why?
   • (v) What is the Confidence Interval for the ratio of variance based of F distribution
2. Q2 Attempt any TWO questions: 20 marks
   • (a) If X follows Gamma(a, A), write its p.d.f. and obtain mean and variance. 10
   • (b) Ifa random variable X follows Beta distribution of type | then state its p.d.f. and Obtain rth order raw moment, mean and variance. (10)
   • (c) State and prove forgetfulness property of Exponential distribution. Suppose the length of minutes) has Exponential distribution with parameter If the shower is already lasted for 3 minutes, what is the probability that it will last for at least two
3. Q3 Attempt any TWO questions: 20 marks
   • (a) i. If then derive expression for even ordered central moments. 8
   • ii. State the Central Limit Theorem. 2
   • (b) i. State five properties of Normal distribution. 5
   • ii. Derive the mode of distribution. 5
   • (c) A test of solved examples was given to 600 candidates. The marks obtained in the test assumed to follow normal distribution with mean 15 and s.d. 16. Find i) how many candidates obtained marks between 13 and 16. ii) how many scored below 12?
   • iii) how many scored ? iv) what is the probability that a candidate chosen at random had scored between 14 and |8? (10)
4. Q4 Attempt any TWO questions: 29 marks
   • (a) i. Derive the distribution of of F 5
   • ii. State the properties of Chi-Square distribution with n degrees of freedom. 5
   • (b) i. Prove that odd ordered moments of t-distribution are equal to zero. 5
   • ii. State the application of Chi- Square distribution. 5
   • (c) i. Two types of batteries are tested for their length of life and the following data are No. of sample __| Mean life(in hrs) Variance Use 5% level of significance, should we conclude that they have same length of life VCD/ SYBSC(MS)-SEM STATISTICS | 100 MARKS 3 HRS
   • ii. A sample of size 10 from a normal population gave the following value Test the hypothesis that population variance is 32. Take 1% level of significance
5. Q5 Attempt any TWO questions: 20 marks
   • (a) i. If r.v. X~U(a. b) with mean=5 , variance=3, obtain a and b. Also find P(X<4). — 5
   • ii. The lifetime of certain device assumed to follow Exponential distribution with Mean 300 hours, What is the probability that such device will last
   • (1) between 150 and 300 hours (2) at least 180 hours. 5
   • (b) Obtain mean and variance of normal distribution with parameters N 10
   • (c) i. State the properties of t- distribution. 5
   • ii. State the properties of F- distribution. 5

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