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12th Standard Maths English Medium Probability Distributions Reduced Syllabus Important Questions 2021

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  1. A rod of length 2l is broken into two pieces at random. The probability density function of the shorter of the two pieces is
    \(f(x)=\left\{\begin{array}{ll} \frac{1}{l} & 0<x<l \\\ 0 & l \leq x<2 l \end{array}\right.\)
    The mean and variance of the shorter of the two pieces are respectively

    (a)

    \(\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 3 } \)

    (b)

    \(\\ \cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 6 } \)

    (c)

    \(l,\cfrac { { l }^{ 2 } }{ 12 } \)

    (d)

    \(\cfrac { l }{ 2 } ,\cfrac { { l }^{ 2 } }{ 12 } \)

  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 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

  3. 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

  4. 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

  5. Two coins are to be flipped. The first coin will land on heads with probability 0.6, the second with probability 0.5. Assume that the results of the flips are independent, and let X equal the total number of heads that result The value of E(X) is

    (a)

    0.11

    (b)

    1.1

    (c)

    11

    (d)

    1

  6. Which of the following is a discrete random variable?
    I. The number of cars crossing a particular signal in a day
    II.The number of customers in a queue-to buy train tickets at a moment.
    III.The time taken to complete a telephone call.

    (a)

    I and II

    (b)

    II only

    (c)

    III only

    (d)

    II and III

  7. If \(f(x)=\left\{\begin{array}{ll} 2 x & 0 \leq x \leq a \\ 0 & \text { otherwise } \end{array}\right.\) is a probability density function of a random variable, then the value of a is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  8. The probability mass function of a random variable is defined as:

    x -2 -1 0 1 2
    f(x) k 2k 3k 4k 5k
    (a)

    \(\cfrac { 1 }{ 15 } \)

    (b)

    \(\cfrac { 1 }{ 10 } \)

    (c)

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

    (d)

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

  9. If \(f(x)={ Cx }^{ 2 }={ cx }^{ 2 },0<x<2\) is the p.d.f, of x then c is

    (a)

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

    (b)

    \(\cfrac { 4 }{ 3 } \)

    (c)

    \(\cfrac { 8 }{ 3 } \)

    (d)

    \(\cfrac { 3 }{ 8 } \)

  10. In eight throws of a die, 1 or 3 is considered a success. Then the mean number of success is

    (a)

    \(\cfrac { 8 }{ 3 } \)

    (b)

    \(\cfrac { 4 }{ 3 } \)

    (c)

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

    (d)

    \(\cfrac { 5 }{ 3 } \)

  11. In a binomial distribution,\(n=4,P(X=0)=\cfrac { 16 }{ 81 } \),then \(P(X=4)\)

    (a)

    \(\cfrac { 1 }{ 16 } \)

    (b)

    \(\cfrac { 1 }{ 81 } \)

    (c)

    \(\cfrac { 1 }{ 27 } \)

    (d)

    \(\cfrac { 1 }{ 8 } \)

  12. If X is a continuous random variable then \(P\left( x\ge a \right) =\) __________.

    (a)

    \(P(X<a)\)

    (b)

    \(1-P(X>a)\)

    (c)

    \(P(X>a)\)

    (d)

    \(1-P(X\le a-1)\)

  13. If F(x) is a distribution function of a random variable then the false statement is

    (a)

    \(F(\infty )=1\)

    (b)

    \(F(-\infty )=-1\)

    (c)

    \({ F }^{ ' }\left( x \right) =f(x)\)

    (d)

    \(0<F(x)<1\)

  14. A random variable X has the following probapality mass function?

    X -2 3 1
    P(X=x) \(\cfrac { \lambda }{ 6 } \) \(\cfrac { \lambda }{ 4 } \) \(\cfrac { \lambda }{ 12 } \)
    (a)

    \(\cfrac { \lambda }{ 6 } +\cfrac { \lambda }{ 4 } +\cfrac { \lambda }{ 12 } =1\)

    (b)

    \(\lambda =2\)

    (c)

    \(P(X=3)=\cfrac { 1 }{ 8 } \)

    (d)

    \(P(1\le X\le 3)=\cfrac { 2 }{ 3 } \)

  15. For a Bernouli distribution

    (a)

    \(\sigma =\sqrt { npq } \)

    (b)

    \(mean=\mu \)

    (c)

    \(\mu =p\)

    (d)

    \({ \sigma }^{ 2 }=pq\)

    1. 2 Marks


    10 x 2 = 20
  16. Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values ofthe random variable X and number of points in its inverse images.

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

  18. Find the probability mass function and cumulative distribution function of number of girl child in families with 4 children, assuming equal probabilities for boys and girls.

  19. A random variable X has the following probability mass function.

    x 1 2 3 4 5
    f(x) k2 2k2 3k2 2k 3k
  20. The probability density function of X is given by \(f(x)=\begin{cases} \begin{matrix} kxe^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & for\quad x\le 0 \end{matrix} \end{cases}\) Find the value of k.

  21. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
    \(f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 3 } & 0<x<30 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Obtain and interpret the expected value of the random variable X .

  22. Compute P(X = k)for the binomial distribution, B(n,p) where
    \(P(X=10)=\left( \begin{matrix} 10 \\ 4 \end{matrix} \right) \left( \cfrac { 1 }{ 5 } \right) ^{ 4 }\left( 1-\cfrac { 1 }{ 5 } \right) ^{ 10-4 }\)

  23. Compute P(X = k)for the binomial distribution, B(n,p) where
    n=9,\(p=\cfrac { 1 }{ 2 } \) ,k=7

  24. Find the binomial distribution function for each of the following.
    (i) Five fair coins are tossed once and X denotes the number of heads.
    (ii) A fair die is rolled 10 times and X denotes the number of times 4 appeared.

  25. Define Continuous random variable.

    1. 3 Marks


    10 x 3 = 30
  26. 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.

  27. 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.

  28. Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by
    \(\\ \\ \\ \\ \\ f(x)=\begin{cases} \begin{matrix} \frac { { x }^{ 2 }+1 }{ k } & forx=0,1,2 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\\ \\ \\ \\ \\ \\ \)
    Find
    (i) the value of k
    (ii) cumulative distribution function
    (iii) P(X ≤ 1).

  29. Four fair coins are tossed once. Find the probability mass function, mean and variance for number of heads occurred.

  30. A lottery with 600 tickets gives one prize of Rs.200, four prizes of noo, and six prizes of Rs. 50. If the ticket costs is Rs.2, find the expected winning amount of a ticket

  31. 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.

  32. 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)

  33. Give any three properties of distribution function.

  34. Find the mean, variance and standard deviation of the number of heads in two tosses of a coin

  35. In a game, a man wins Rs.5 for getting a number greater than 4 and loses Re.1 otherwise when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

    1. 5 Marks


    7 x 5 = 35
  36. The probability density function of X is given
    \(f(x)=\begin{cases} \begin{matrix} { Ke }^{ \frac { -x }{ 3 } } & \begin{matrix} for & x>0 \end{matrix} \end{matrix} \\ \begin{matrix} 0 & \begin{matrix} for & x\le 0 \end{matrix} \end{matrix} \end{cases}\)
    Find
    (i) the value of k
    (ii) the distribution function.
    (iii) P(X <3)
    (iv) P(5 ≤X)
    (v) P(X ≤ 4)

  37. If X is the random variable with probability density functionj{x) given by,
    \(f(x)=\begin{cases} \begin{matrix} x+1 & -1\le x<0 \end{matrix} \\ \begin{matrix} -x+1 & 0\le x<1 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
    then find
    (i) the distribution function F(x)
    (ii) P( -0.5 ≤X ≤ 0.5)

  38. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
    (i) exactly 10 will have a useful life of at least 600 hours;
    (ii) at least 11 will have a useful life of at I least 600 hours; I
    (iii) at least 2 will not have a useful life of at : least 600 hours.

  39. 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 ) .

  40. Let X be a random variable denoting the life time of an electrical equipment having probability density function
    \(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\) 
    Find
    (i) the value of k
    (ii) Distribution function 
    (iii) P(X < 2)
    (iv) calculate the probability that X is at least for four unit of time 
    (v) P(X = 3)

  41. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.

  42. If X is a binomial random variable with mean 4 and variance 2 find \(P(|X-2|\le 2)\)

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