Statistics and Probability Answers

6. (a) For normal distribution with mean zero and variance 2

σ show that:

2

(| |) σ

π

E x =

 (

5(b) For the given bivariate probability distribution of X and Y : (5) 

32

( , )

2

x y

P X x Y y

+

= = = for x = 3,2,1,0 and y = .1,0

 Find: 

 (i) P(X ≤ ,1 Y = )1

 (ii) P(X ≤ )1

 (iii) P(Y > )0 and 

 (iv) P(Y = |1 X = )3 

5. (a) Suppose X is a gamma variate with E(x) = 3 and var(X ) = .7 Find the parameters 

α and λ of the gamma distribution. 

4. (a) Five unbiased dice were thrown 96 times and the number of times 4, 5 or 6 was 

obtained, is given in the following table: (5) 

No. of dice 

showing 4, 

5 or 6 

0 1 2 3 4 5

Frequency 1 10 24 35 18 8

At 5% level of significance test whether this data comes from a binomial distribution. 

You may like to use the following values. 

 [( 10 05. ) 11 07. ,

2

x5 = 05.0( ) 12 59. ,

2

x6 = 05.0( ) 14 07. .] 2

x7 = 

 (b) The yield (in kg) of 100 plots in the form of grouped frequency distribution is given 

below: (5) 

Yield 

(kg) 

Frequency

0-20 6 

20-40 21 

40-60 35 

60-80 30 

80-100 8 

(i) Estimate the number of plots with an yield of 

(A)40 to 80 kg 

(B) 10 to 70 kg 

 (ii) Find the mean and standard deviation of yield.

(b) The yield (in kg) of 100 plots in the form of grouped frequency distribution is given 

below: (5) 

Yield 

(kg) 

Frequency

0-20 6 

20-40 21 

40-60 35 

60-80 30 

80-100 8 

(i) Estimate the number of plots with an yield of 

(A)40 to 80 kg 

(B) 10 to 70 kg 

 (ii) Find the mean and standard deviation of yield.

4. (a) Five unbiased dice were thrown 96 times and the number of times 4, 5 or 6 was 

obtained, is given in the following table: (5) 

No. of dice 

showing 4, 

5 or 6 

0 1 2 3 4 5

Frequency 1 10 24 35 18 8

At 5% level of significance test whether this data comes from a binomial distribution. 

You may like to use the following values. 

 [( 10 05. ) 11 07. ,

2

x5 = 05.0( ) 12 59. ,

2

x6 = 05.0( ) 14 07. .]

3) (a) If the moment generating function (m.g.f.) of a random variable X is 

( ) exp 3( 32 ). 2 M t t t X = + Find mean and standard derivation of X and also compute 

P(x < ).3 (6) 

 (b) The probability density function of a random variable X is f (x) = C | x |; Find C,

and the value of 0

x such that ,

4

3

( ) FX

x0 = where F is the CDF.

2) A,B and C are three events. Express the following events in set notations. (10) 

 (i) Simultaneous occurrence of A,B and C.

 (ii) Occurrence of at least one of them. 

 (iii) Both A and B occur and C does not occur. 

 (iv) The event B but not A occurs. 

 (v) Not more than one of the events A,B and C occur.

1. State whether the following statements are true or false. Give a short proof or a counter 

example in support of your answers: (10) 

 (a) Poisson distribution is a limiting case of binomial distribution for n p, 1

→ → ∞ and 

np ∞.

→

 (b) For two independent events A and B, if P(A) = 2.0 and P(B) = ,4.0 then 

(A∩ B) = .6.0

 (c) If H0

: P ≤ 6.0 and X ~ B(n, p)n -known and p unknown and 1 0 H :µ = µ where 

X ~ N

2 2

(µ,σ )σ unknown, then H0

 and H1

 are simple null hypothesis. 

 (d) Frequency density of a class for any distribution is the ration of total frequency to 

class width. 

 (e) If X and Y are independent r.v.s with M (t) X

 and M (t) Y

 as their m.gf’s 

respectively, then M (t) M (t)M t)

The probability that a student is accepted to a prestigious college is 30%. If 5 students from the same school apply: What is the expected number of students accepted?