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Probability Distributions 5 Mark Book Back Question Paper With Answer Key

12th Standard

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Maths

Time : 01:00:00 Hrs
Total Marks : 275

     5 Marks

    55 x 5 = 275
  1. In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images.

  2. An urn contains 5 mangoes and 4 apples. Three fruits are taken at randaom. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

  3. 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.

  4. 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.

  5. 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)

  6. 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.

  7. 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).

  8. 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)

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

    x 1 2 3 4 5
    f(x) k2 2k2 3k2 2k 3k

    Find
    (i) the value of k
    (ii) P(2 \(\le\) X < 5)
    (iii) P(3 < X )

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

    Find
    (i) the probability mass function
    (ii) P(X < 3) and
    (iii) P(X \(\ge \)2).

  11. The probability density function of X is 
    \(f(x)=\left\{\begin{array}{cc} x & 0
    find P(0.2 ≤ X< 0.6) 

  12. The probability density function of X is 
    \(f(x)=\left\{\begin{array}{cc} x & 0
    find P(1.2 ≤ X < 1.8) 

  13. The probability density function of X is 
    \(f(x)=\left\{\begin{array}{cc} x & 0
    find P(0.5≤X<1.5)

  14. Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function 
    \(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
    Find
    (i) the value of k
    (ii) the distribution function
    (iii) the probability that daily sales will fall between 300 litres and 500 litres?

  15. 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)

  16. If X is the random variable with probability density function f(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)

  17. If X is the random variable with distribution function F(x) given by,

    then find (i) the probability density function f(x) 
    (ii) P(0.3 ≤ X ≤ 0.6)

  18. For the random variable X with the given probability mass function as below, find the mean and variance.
    \(f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 10 } & x=2,5 \end{matrix} \\ \begin{matrix} \frac { 1 }{ 5 } & x=0,1,2,3,4 \end{matrix} \end{cases}\) 

  19. For the random variable X with the given probability mass function as below, find the mean and variance 

  20. For the random variable X with the given probability mass function as below, find the mean and variance \(f(x)= \begin{cases}2(x-1) & 1

  21. For the random variable X with the given probability mass function as below, find the mean and variance.
    \(f(x)=\begin{cases} \begin{matrix} \cfrac { 1 }{ 2 } e^{ -\frac { x }{ 2 } } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)

  22. Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X.

  23. If μ and σ2 are the mean and variance of the discrete random variable X, and E(X + 3) =10 and E(X + 3)2 = 116, find μ and \(\sigma\)2

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

  25. 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}\frac{1}{30} & 0

  26. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function \(f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Find the expected life of this electronic equipment.

  27. The probability density function random variable X is given by \(f(x)=\begin{cases} \begin{matrix} { 16xe }^{ -4x } & for\quad x>0 \end{matrix} \\ \begin{matrix} 0 & for\quad x\le 0 \end{matrix} \end{cases}\) find the mean and variance of X.

  28. A lottery with 600 tickets gives one prize of Rs. 200, four prizes of Rs.100, and six prizes of Rs. 50. If the ticket costs is Rs. 2, find the expected winning amount of a ticket.

  29. Compute P(X = k) for the binomial distribution, B(n, p) where
    n = 6, \(p=\frac { 1 }{ 3 } \), k = 3

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

  31. The probability that Mr.Q hits a target at any trial is \(\frac { 1 }{ 4 } \). Suppose he tries at the target 10 times. Find the probability that he hits the target
    (i) exactly 4 times
    (ii) at least one time.

  32. Using binomial distribution find the mean and variance of X for the following experiments
    (i) A fair coin is tossed 100 times, and X denote the number of heads.
    (ii) A fair die is tossed 240 times, and X denote the number of times that four appeared.

  33. The probability that a certain kind of component will survive a electrical test is \(\frac { 3 }{ 4 } \). Find the probability that exactly 3 of the 5 components tested survive.

  34. A retailer purchases a certain kind of electronic device from a manufacturer. The manufacturer, indicates that the defective rate of the device is 5%. The inspector of the retailer randomly picks 10 items from a shipment. What is the probability that there will be
    (i) at least one defective item
    (ii) exactly two defective items.

  35. 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 least 600 hours
    (iii) at least 2 will not have a useful life of at least 600 hours.

  36. The mean and standard deviation of a binomial variate X are respectively 6 and 2.
    Find
    (i) the probability mass function
    (ii) P(X = 3)
    (iii) P(X\(\ge \)2).

  37. If X~ B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6. Find the distribution, mean and standard deviation of X.

  38. In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the mean and variance of the random variables.

  39. 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.

  40. A pair of fair dice is rolled once. Find the probability mass function to get the number of fours.

  41. If the probability mass function f(x) of a random variable X is

    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)

  42. A six sided die is marked ‘1’ on one face, ‘2’ on two of its faces, and ‘3’ on remaining three faces. The die is rolled twice. If X denotes the total score in two throws.
    (i) Find the probability mass function.
    (ii) Find the cumulative distribution function.
    (iii) Find P(3 ≤ X< 6)
    (iv) Find P(X ≥ 4) .

  43. 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(\(X \geq-1)\)

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

    x  1   2  3  4  5  6
    f(x)  k  2k   6k   5k   6k   10k 

    Find
    (i) P(2 < X < 6)
    (ii) P(2 ≤ X < 5)
    (iii) P(X ≤4)
    (iv) P(3 < X )

  45. 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 )

  46. If X is the random variable with probability density function f(x) given by,

    \(f(x)=\begin{cases} \begin{matrix} x-1 & 1\le x<2 \end{matrix} \\ \begin{matrix} -x+3 & 2\le x<3 \end{matrix} \\ \begin{matrix} 0 & Otherwise \end{matrix} \end{cases}\) 
    find
    (i) the distribution function F(x)
    (ii) P(1.5 ≤ X ≤ 2.5)

  47. 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)

  48. The probability density function of random variable X is given by \(f(x)=\begin{cases} \begin{matrix} k & 1\le x\le 5 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) Find
    (i) Distribution function
    (ii) P(X < 3)
    (iii) P(2 < X < 4)
    (iv) P(3 ≤ X )

  49. 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)

  50. Suppose that f (x) given below represents a probability mass function

    x 1 2 3 4 5 6
    f(x) c2 2c2 3c2 4c2 c 2c

    Find
    (i) the value of c
    (ii) Mean and variance.

  51. 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 

  52. 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.

  53. A multiple choice examination has ten questions, each question has four distractors with exactly one correct answer. Suppose a student answers by guessing and if X denotes the number of correct answers, find
    (i) binomial distribution
    (ii) probability that the student will get seven correct answers
    (iii) the probability of getting at least one correct answer

  54. The mean and variance of a binomial variate X are respectively 2 and 1.5. Find 
    (i) P(X = 0)
    (ii) P(X =1)
    (iii) P(X ≥1)

  55. On the average, 20% of the products manufactured by ABC Company are found to be defective. If we select 6 of these products at random and X denote the number of defective products find the probability that
    (i) two products are defective
    (ii) at most one product is defective
    (iii) at least two products are defective.

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