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    Code No: 21BS1MT01 HR-22

    HYDERABAD INSTITUTE OF TECHNOLOGY & MANAGEMENT


    (AUTONOMOUS)
    B. Tech II Year I Semester Examinations, Feb-2024
    Statistical and Mathematical Foundations
    (Common to CSC, CSD, CSM, CSO)
    Time: 3 Hours Max Marks: 60

    Note: 1. Question paper contains Part-A and Part-B


    2. Part-A carries 10 Marks. Answer all Questions
    3. Part-B consists of 10 Questions. Answer FIVE questions, each question carries
    10 marks may have a, b, c as sub questions.

    PART-A: (10 x 2 marks = 20)


    1a) A random variable X has probability density function f(x) as given below. 𝑓(𝑥) = [2]
    𝑎 + 𝑏𝑥 𝑓𝑜𝑟 0 < 𝑥 < 1
    { If the expected value E[X] = 2/3, then find
    0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
    Pr[X<0.5]
    b) The random variable X takes on the values 1, 2 (or) 3 with probability [2]
    2+5𝑃 1+3𝑃 1.5+2𝑃
    5
    , 5 𝑎𝑛𝑑 5 respectively the values of P and E(X) are respectively.

    c) 2
    Suppose that X has a Poisson distribution. If 𝑃(𝑋 = 2) = 3 𝑃(𝑋 = 1). Evaluate [2]
    P(x=0).
    d) If X is a continuous random variable whose probability density function is given [2]
    by
    𝑘(5𝑥 − 2𝑥 2 ), 0 ≤ 𝑥 ≤ 2
    f(x) = { . Then P(x>1) is
    0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
    e) The joint probability density function of two variables (X, Y) is given by [2]
    𝑘𝑒 −(𝑥+𝑦) : 0<𝑦≤𝑥<∞
    f(x,y) = { then find K.
    0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
    f) The uniform random variable has mean 1.5 and variable 27/4, then find [2]
    P(X>0).

    g) Find the value of the finite population correction factor for the size of sample [2]
    10 and the size of population 1000.

    h) Find the maximum error so that one can expect to make with the probability [2]
    0.90 when using the mean of a random sample of size 64 to estimate the mean
    of population with variance 2.56.
    i) Using Fermat factorization find the factor of the integer 11021. [2]

    j) Find GCD and LCM of 720 and 2100. [2]


    PART-B: ( 5 x 10 marks = 50)
    2) Machines A, B and C all produce the same two parts, X and Y. Of all the parts [10]
    produced, machine A produces 60%, machine B produces 30%, and machine
    C produces 10%. In addition, 40% of the parts made by machine A are part
    X. 50% of the parts made by machine B are part X. 70% of the parts made by
    machine C are part X. A part produced by this company is randomly sampled
    and is determined to be an X part. With the knowledge that it is an X part,
    revise the probabilities that the part came from machine A
    (OR)
    3) Two dice are thrown. Let X assign to each point (a, b) in S the maximum of [10]
    its number i.e., X(a, b) = max(a, b). find the probability distribution. Also find
    mean and variance of the distribution

    4) a) 20% of items produced from a factory are defective. Find the probability [5+5]
    that in a sample of 5 chosen at random, i) none is defective, ii) one is
    defective, iii) P(1<X<4).
    b) If the variance of Poisson variable is 1.8, then find i) P(X>1) ii) P(X=5) iii)
    P(0<X<5).

    (OR)
    5) In a distribution exactly normal, 10.03% of the items are under 25-kilogram [10]
    weight and 89.97% of the items are under 70-kilogram weight. What are the
    mean and Standard deviation of the distribution?

    6) The joint density function of the random variables (x, y) is given as follows [10]
    2
    𝑓(𝑥, 𝑦) = {𝑐𝑥 𝑦 ∶ 0 ≤ 𝑥, 𝑦 ≤ 1 find i) Cov (x, y) ii) E[X+Y]
    0 ∶ 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

    (OR)
    7) a) Calculate the rank coefficient of correlation for the following data. [10]
    X 50 50 55 60 65 65 65 60 60 60
    Y 11 13 14 16 16 15 15 14 13 13

    8) A company claims that its bulbs are superior to those of its main competitor. [5+5]
    If a study showed that a sample of 40 bulbs have a mean life time of 647hrs
    of continuous use with standard deviation of 27 hrs. while a sample of 40
    bulbs made by its main competitor had a mean life time of 638 hrs of
    continuous use with standard deviation of 31 hrs. test the significance
    difference between the difference of means at 5% level.
    (OR)
    9) A population consists of 5 numbers 2, 6, 8, 11, 12. Consider all the samples of [10]
    size 2 that can be taken from the population with replacement, then find
    i) Mean of Population, ii) Standard Deviation of the population, iii) Mean of
    sampling distribution of means, iv) Standard Deviation of sampling
    distribution of means, v) Standard error.
    10) a) find the solution of system of congruences 𝑥 ≡ 1 (𝑚𝑜𝑑 2), 𝑥 ≡ [5+5]
    2 (𝑚𝑜𝑑 3), 𝑥 ≡ 3 (𝑚𝑜𝑑 5).
    b) Solve the linear congruences 9𝑥 ≡ 21 (𝑚𝑜𝑑 30).
    (OR)
    11) a) Using Euclidean algorithm calculate g.c.d. (12378, 3054) and obtain [10]
    integers x and y satisfying g.c.d. (12378, 3054) = 12378x+3054y.

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