ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern): Questions 28 - 32 of 164

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Question 28

Appeared in Year: 2011

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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

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Appeared in Year: 2014

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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&#8230;

… (13 more words) …

Question 30

Appeared in Year: 2013

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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

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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

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Appeared in Year: 2012

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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&#8230;

… (11 more words) …