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Probability Distributions Five Marks Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 45
    9 x 5 = 45
  1. Two fair coins are tossed simultaneously (equivalent to a fair coin is tossed twice). Find the probability mass function for number of heads occurred.
     

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

  3. 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)

  4. 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) .

  5. 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)\(P(X\ge -1\) 

  6. 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 )

  7. 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 )

  8. 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 Rs10 for each white ball selected. Find the expected winning amount and variance 

  9. Find the mean and variance of a random variable X , whose probability density function is \(f(x)=\begin{cases} \begin{matrix} { \lambda e }^{ -2x } & for\ge 0 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\)

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