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Probability Distributions Model Question Paper

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
    5 x 1 = 5
  1. Consider a game where the player tosses a sixsided fair die. H 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 = {I, 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)

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

  2. 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 roUed 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

  3. 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

  4. If P{X = 0} = 1- P{X = I}. IfE[X) = 3Var(X), then P{X = 0}.

    (a)

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

    (b)

    \(\cfrac { 2 }{ 5 } \)

    (c)

    \(\cfrac { 1 }{ 5 } \)

    (d)

    \(\cfrac { 1 }{ 3 } \)

  5. Suppose that X takes on one of the values 0, 1, and 2. If for some constant k, P(X = I) = k P(X = i-I) i = 1, 2 and P(X = 0) =\(\cfrac { 1 }{ 7 } \) then the value of k is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  6. 3 x 2 = 6
  7. 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.

  8. Three fair coins are tossed simultaneously. Find the probability mass function for number of heads occurred

  9. A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
    (i) the probability mass function
    (ii) the cumulative distribution function
    (iii) P(4 ≤ X < 10)
    (iv) P(X ≥ 6)

  10. 3 x 3 = 9
  11. 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.

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

  13. 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)

  14. 2 x 5 = 10
  15. If the probability mass function f (x) of a random variable X isx

    x 1 2 3 4
    f (x) \(\cfrac { 1 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 5 }{ 12 } \) \(\cfrac { 1 }{ 12 } \)

    find (i) its cumulative distribution function, hence find
    (ii) P(X ≤ 3) and,
    (iii) P(X ≥ 2)

  16. Find the probability mass function f (x) of the discrete random variable X whose cumulative distribution function F(x) is given by

    Also find (i) P(X < 0) and (ii)\(P(X\ge -1\) 

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