BSc Statistics SEM III 2022 2023 Oct 2023 STATISTICS I Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Answer the following questions
   • a) Correct the following if necessary: (10M)
   • i. M.G.F. uniquely determines the distribution
   • ii. Binomial distribution is always unimodal
   • iii. All Cumulants are unequal in case of Poisson distribution
   • iv. distribution has 2 parameters
   • v. If E(Y/X) = E(Y) then X and Y are independent
   • b) Answer in One sentence: (10M)
   • i. Write down p.m.f. of discrete uniform variate over the range {1,2,.,.n} and also state its mean and variance
   • ii. Ifthe moment generating function is given by My(t)= [0.55 + Write down its probability mass function and mean
   • iii. | Write down the p.m.f of poison variable and write the expression fer mean of
   • iv. Derive the expression for M.G.F. of Geometric Distribution
   • v. Define conditional probability density of X for given values of Y
2. Q2 Attempt any TWO (20M)
   • a) with M.G.F. My(t). Then prove that 07
   • (i) M.G.F. of Y = X+b is My(t) =
   • (ii) M.G.F. of Y = aX+b is My(t) =
   • (iii) If X and Y are two independent variables with their respective M.G.F. My(t) and My(t) , then M.G.F. of X+Y is = My(t) (iv)
   • (2) If r.v. X has 03 VCD/ SEM-III STATISTICS-I 100 MARKS Show that its M.G.F. is given by Obtain its M.G.F, Find E(X) and V(X). State M.G.F. of Find E(Y)
   • c) Ifarandom variable X follows Bernoulli Distribution with probability p Obtain expression for its moment generating functiou. Hence evaluate its mean, variance and measure of skewness
3. Q3 Attempt any TWO (20M)
   • a) (i) State and prove the Memory Loss Property for Geometric Distribution with
   • (ii) Ifa random variable X follows Poisson distribution if P(X = 5) = P(X = 4) Then find SD and mode. U3
   • b) Prove that the sum of two independent Poisson variates is a Poisson variate while the difference is not a Poisson variate
   • c) Evaluate the mean and variance using for Geometric function
4. Q4 Attempt any TWO (20M)
   • a) Define marginal and conditional probability mass function .derive conditional : b) (i) State and prove additional theorem on Expectation of two discrete
   • (ii) Multiplication theorem on Expectation of two discrete random variables, SEM-III STATISTICS-I 100 MARKS
   • c) For the following joint p.m.f of X, Y
   • (i) Examine whether X and Y are independent?
   • (ii) Find P[X+Y<3]
5. Q5 Attempt any TWO (20M)
   • a) (i) With usual notations, if py(t) = (2 — Write its M.G.F. Find E(X)
   • (ii) For a binomial variate mean is 6 and variance is 4, find (2)P(X>2)
   • b) The number of monthly breakdown of a computer is a random variable having Poisson distribution with mean 1.8. Find the probability that this computer will work for a month
   • i) with only one defective
   • ii) With at least 2 defective
   • c) If X and Y are two discrete random variables with E(X) = 10, V(X) =20, V(Y) = 16 and Cov(X,Y) = 2 Obtain: (i) E(4X + 2Y +5), (ii) V(3X + 2Y), (iii) — Y +2),
   • (iv) Cov(2X + 1, + 2)

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