HSC Statistics 2nd Paper Model Test-1

Class – XII

Subject Statistics 2nd Paper

Time : 3 hrs.                                                                                                                                         Marks.75

 N.B. The figures in the right margin indicate full marks.)

1.         a)         Define with example.

                        (i)         Simple event                             (ii)        Complementary event.

                        (iii)       Sure event                                (iv)       Impossible event

            b)         Define conditional event. State and prove the multiplicative law of probability for two

            dependent event.

            c)         A box contain 5 red,  2 white and 4 black balls. 3 balls are drawn at random. What is

            the probability that  (i)  balls are same colour  (ii)  At best two balls are black.

Or,       a)         Define                                                                                                                                      4

                        (i)         Probability density function.      

                        (ii)        Distribution function.

                        (iii)       Joint probability function.

                        (iv)       Marginal probability function.

            b)         A random variable x is follow the probability function. Find its mean and variance.          4

                        P (x)     =           x,  2. 3 ...........10.

                                    =  0   otherwise.

            c)         What do you mean by mathematical expectation of a random variable? State and prove the             multiplicative law of expectation for two independent random variables.                                                 5

2.         a)         Define Binomial distribution.                                                                                                      2

            b)         Write down the assumption of Binomial distribution.                                                      3

            c)         Calculate mean and variance of a binomial. Distribution and relation between them.        7

            d)         If the probability of a boy or a girl is same then what is the probability that out of                        3

                        5 children there is          (i)  at least one boy.

Or,       a)         Define poison variable                                                                                                   2

            b)         Under certain conditions derive the probability function of poisson distribution.               5

            c)         Write down the properties of poisson distribution.                                                                     3

            d)         In a poisson distribution if P(x=2) =  P (x =4) the find the value of P(x=0) and P(x ³1)           3

            e)         Write down 5 examples of poison distribution.                                                               2

3.         a)         Define standard normal variate. Show that the mean and variance of a standard normal

            variety are 0 and 1 respectively.                                                                                                            1+4

            b)         What is normal distribution? Write down the probability density function of normal

            distribution and express its different symbols.                                                                             1+3

            c)         Write down the importance and properties of normal distribution.                                              6 

Or,       a)         What do you mean by cost of living index number? Discuss its calculation method.

            Write down the bases of index number.                                                                            2+5+2=9

            b)         What do you mean by census and sample survey. Write the advantages of sample     2+4= 6

            survey. 

4.         a)         Define and distinguish between apriori and posteriori probability.                                        2+3=5

Or,       3 cards are taken at random from a pack of 52 cards. What is the probability that.                       2.5+2.5=5

            (i)    2  aces

            (ii)  2 spade.

5.         Show that,        (i)         E (x²) ³ {E (x) }²                                                                                  2.5+2.5=5

                                    (ii)        E (2x+3) = 2E(x) +3.

Or,       The P. d. f.  of a continuous random variety is given below.  f (x) = K  x  0 £ x £ 2

                        (i)         Find the value of k.                                                                                           2+3 =5

             at        (ii)        P ( 0 £ x £ 1).

6.         Derive the recursion formula for Binomial distribution.                                                                5

Or,       The mean and s.d of a Binomial distribution are 16 and   respectively. find its parameter

            and  P (x £ 1 )                                                                                                                            2+3=5

7.         Write down the properties of expectation.                                                                                 5

Or,       Write down the properties of variance.                                                                          5

8.         What do you mean by distribution function? Write down the properties of distribution function.

Or,       Mean and variance of a random variable x are 48 and 36 respectively. Find the probability of

            (i)  P (x ³ 50)               (ii)  P (42 £  x  £  54)                                                                           2+3=5

9.         How can the time of population being doubled be determined.                                                   5

Or,       In a bulb factory. Three machines A. B. C produces 25% and 40% respectively of the production. The product of the machines are respectively 5%,  4% and 2% are defective. From the bulb produced one is taken at random and is found to he defective. Find the probability that the bulb may be a product of machine.