BSc Data Science SEM II 2021 2022 May 2022 DS PROBABILITY AND DISTRIBUTION Question Paper - Mumbai University | munotes

Questions asked in this paper

1. Q1 Attempt the following 40 marks
2. Q1 A variable that can assume any possible value between two points is called
3. Q2 The probability function of a random variable is defined as: -2 0.1 2, f(x):k 2k 3k 4k Sk, Then k is equal to:
   • a) Zero b) 1/4 e)I/IS d) One
4. Q3 The area under the curve of a continuous probability density function is always =
5. Q4 The formula of variance of uniform distribution is
6. Q5 The p.d.f. of normal distribution is an important role in the theory of and Survival analysis and Queuing theory
7. Q7 follows a Gamma distribution with (a, 1) then it is a probability density of ii
8. Q8 Ifrandom variable X follows Gamma (a =2 , then the valuc of its
9. Q9 Area to the right of mean under the normal curve
10. Q10 St of all possible outcomes is known as !1) Two events A and B are mutually exclusive events if
11. Q12 A and B are mutually exclusive events then P(A U B) =
    • a) Zero
    • a) P(B) d) Not fixed
12. Q14 A veriable which can assume finite or countably infinite number of values is known as
13. Q15 For Probability mass function, following condition /conditions s itisfied Which of the following statement is true for cumulative distribution function F(x)?
    • a) F(x) is decreasing function b) F(x) ranges from 0
    • c) F(x) is non-decreasing function d)P(a<x< b) =F (b) — F (a) — P (b)
14. Q17 For Probability density function f(x), following condition /conditions
    • a) x 0) b) f@)dx = 1
    • c) f (x) = 0,V x 0) and =1 d)f(x)=0,Vx or =1
15. Q18 For discrete random variable, the expected value E(x) =
16. Q19 The characteristic function of random variable X is
    • a) E(X)+E(Y) E(X) E(Y) d) ab E(X) E(Y)
17. Q2 A) Attempt the following (Solve any 01)
18. Q1 Define the following terms with suitable example
19. Q2 Prove the following statement
    • a) P (A’) = 1- P (A), where A’ is complementary event of A
    • b) If A and B are mutually exclusive events then P (AUB) = P (A) + P (B)
    • B) Attempt the following (Solve any 01) MARKS}
20. Q1 5 Indians and 3 Americans stand for photograph randomly. Find the probability that
    • a) Two positions are occupied by Indians. b) American are ail adjacent
21. Q2 For two events A and B , if P(A) = 0.98 and P(AUB) = 0.9 ,then find P(B)
    • a) If A and B are independent. A and B are mutually exclusive
22. Q3 A) Attempt the following (Solve any 01) [ 4 marks
23. Q1 A random variable X has the following probability distribution with p.m.f. p(X) ‘a) Find the value of k
    • b) Find P(X <3)
24. Q2 Verify function is probability density function ‘
    • B) Attempt the following (Solve any 01) [03
25. Q1 Define the following
    • a) Random variable. b) Discrete random variable with suitable example
    • c) Continuous random varial:le with suitable example
26. Q2 Define Cumulative distribution function for continuous random variable Also state the properties of Cumulative distribution function
27. Q4 “A) Attempt the following (Solve any 01) [04
28. Q1 Define Mathematical Expectation of discrete random variable prove that
    • a) E(aX+b) = a E(X) +b b) E(b) =b
29. Q2 variable X has probability mass function as, Then obtain Moment generating function. Also find mean
    • B) Attempt the following (Solve any 01) ) If the bivariate probability mass function is given by Then find a) value of marginal pmf of X 3
30. Q2 Define Characteristic function. State all its properties
31. Q5 A) Attempt the following (Solve any 01) [04 ) Derive mean of binomial distribution. Also find recurrence relations of probabilities of
32. Q2 Define hypergeometric distribution. Also find it’s mean
    • B) Attempt the following (Solve any 01) [03 MARKS} ) For Poisson variate X, = P(X = 2). Find P(X=4)
33. Q2 In laboratory there are 6 non-defective computers and 4 defective computers. A random sample of 5 computers is selected state the probability distribution of X, the number of defectives in sample. Find the probability that the sample contains at least one defective computer
34. Q6 A) Attempt the following (Solve any 01) [04 ) Define rectangular distribution. Also write it’s cumulative distribution function And draw it’s graph of pdf and cdf
35. Q2 Define normal distribution. Also write its properties briefly
    • B) Attempt the foiluwing (Solve any 01) [ ) Define standard normal distribution. If Z~N (0, 1) find P (Z<1) for P (Z=1) = 0.3413 3
36. Q2 The waiting time are a bus stop is assured to follow rectangular distribution over (5, 15) What is the chance that a person arriving at bus stop gets bus Between 8 to min

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