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Probability Distributions Model Question Paper

12th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 35
    5 x 1 = 5
  1. Let X be random variable with probability density function
    \(f(x)=\left\{\begin{array}{ll} \frac{2}{x^{3}} & x \geq 1 \\ 0 & x<1 \end{array}\right.\)
    Which of the following statement is correct 

    (a)

    both mean and variance exist

    (b)

    mean exists but variance does not exist

    (c)

    both mean and variance do not exist

    (d)

    variance exists but Mean does not exist

  2. Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins Rs. 36, otherwise he loses Rs. k2, where k is the face that comes up k = {1, 2, 3, 4, 5}.
    The expected amount to win at this game in Rs. is

    (a)

    \(\cfrac { 19 }{ 6 } \)

    (b)

    \(-\cfrac { 19 }{ 6 } \)

    (c)

    \(\cfrac { 3 }{ 2 } \)

    (d)

    \(-\cfrac { 3 }{ 2 } \)

  3. A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  4. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

    (a)

    6

    (b)

    4

    (c)

    3

    (d)

    2

  5. Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are

    (a)

    50,40

    (b)

    40,50

    (c)

    40.75,40

    (d)

    41,41

  6. 4 x 2 = 8
  7. Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values of the random variable X and number of points in its reverse images.

  8. Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win Rs. 15 for each red ball selected and we lose Rs. 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images.

  9. A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.

  10. The cumulative distribution function of a discrete random variable is given by

    Find
    (i) the probability mass function
    (ii) P(X < 1 ) and
    (iii) P(X \(\geq\)2)

  11. 4 x 3 = 12
  12. Suppose a pair of unbiased dice is rolled once. If X denotes the total score of two dice, write down
    (i) the sample space
    (ii) the values taken by the random variable X,
    (iii) the inverse image of 10, and
    (iv) the number of elements in inverse image of X.

  13. Two balls are chosen randomly from an urn containing 6 white and 4 black balls. Suppose that we win Rs. 30 for each black ball selected and we lose Rs. 20 for each white ball selected. If X denotes the winning amount, then find the values of X and number of points in its inverse images.

  14. Find the constant C such that the function \(f(x)= \begin{cases}C x^{2} & 1 is a density function, and compute
    (i) P(1.5 < X < 3.5)
    (ii) P(X ≤ 2)
    (iii) P(3 < X )

  15. If X is the random variable with distribution function F(x) given by,
    \(F(x)=\begin{cases} \begin{matrix} 0 & x<0 \end{matrix} \\ \begin{matrix} x & 0\le x<1 \end{matrix} \\ \begin{matrix} 1 & 1\le x \end{matrix} \end{cases}\) 
    then find
    (i) the probability density function f(x)
    (ii) P(0.2 ≤ X ≤ 0.7)

  16. 2 x 5 = 10
  17. A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.

  18. Two balls are chosen randomly from an urn containing 8 white and 4 black balls. Suppose that we win Rs. 20 for each black ball selected and we lose Rs. 10 for each white ball selected. Find the expected winning amount and variance 

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