ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 28 - 32 of 164
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Question 28
Appeared in Year: 2011
Describe in Detail
Essay▾Let Z be a random variable with p. d. f. f (z) . Let zα be its upper α th quantile. Show that if X is
a random variable with p. d. f. then σzα + µ is the upper α th quantile of X.
Explanation
Let Z be a random variable with p. d. f. f (z) . Given that the quantile of z is defined as
X is a random variable with p. d. f. , then quantile is
Assume
Then the limit is
Lower limit is - and upper limit is
If z = then the integral is same for given z and the quantile is
So, the quantile of X is .
Question 29
Appeared in Year: 2014
Describe in Detail
Essay▾Show that for 40,000 throws of a balanced coin, the probability is at least 0·99 that the proportion of heads will fall between 0·475 and 0·525.
Explanation
For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40,000, the mean and standard deviation is
Using Chebyshev՚s inequality,
The question says that the probability is at least 0.99 that is
So, the number of heads comes between is 19000 to 21000.
Hence, the probability is at least 0.99 that the proportion of heads will fall bet…
… (13 more words) …
Question 30
Appeared in Year: 2013
Describe in Detail
Essay▾Use Simpson՚s one-third rule to estimate approximately the area of the cross section of a river 80 feet wide, the depth d (in feet) at a distance x from one bank being given by the following table. :
x | 0 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
d | 0 | 4 | 7 | 9 | 12 | 15 | 14 | 8 | 3 |
Explanation
T he one-third rule of the integrand using Simpson՚s rule is
The area of the cross section of a river 80 feet wide is
Here n = 8, b = 80, a = 0, h = (b-a) /n = 10
… (2 more words) …
Question 31
Appeared in Year: 2011
Describe in Detail
Essay▾Let X1 , X2 , … , Xn be a sequence of i. i. d. r. v. s with E (Xi) = 0 and V (Xi) = 1. Show that the sequence tends to 1 in probability.
Explanation
we have known that is the sample variance of the sequence. The mean is
The convergence in probability is
Using Chebychev՚s inequality
Thus, sufficient condition is that convergence in probability to 1 is that .
… (2 more words) …
Question 32
Appeared in Year: 2012
Describe in Detail
Essay▾The ith box contains 2i white balls and 6 - 2i black balls, i = 1 (1) 3. A fair die is cast once. 3 balls are taken at random from box 1, box 2 or box 3 according as the die shows up face 1 , any of 2 and 3, or any of 4,5 and 6, respectively. Let X denotes the number of white balls drawn. Find E (X) .
Explanation
E1 = Box 1 2 white and 4 black when the fair dice value is x1 = 1
E2 = Box 2 4 white and 2 black when the fair dice value is x2 = (2,3)
E3 = Box 3 6 white when the fair dice value is x3 = (4,5, 6)
X denotes the number of white balls out of randomly select 3 balls
Probability of selecting a ball in E1 is
Similarly, Probability of selecting a ball in E2 i…
… (11 more words) …