Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 64 - 71 of 72
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Question number: 64
» Probability » Standard Probability Distributions » Poisson
Appeared in Year: 2011
Describe in Detail
Show that the sum of two independent Poisson random variables with parameters λ and µ respectively is a Poisson random variable with parameter λ+µ.
Explanation
Let X and Y are independent Poisson random variables with parameters λ and µ respectively. We proof this by moment generating function. The moment generating function of Poisson distribution is
So, the sum of X and Y moment generating function is
because X and Y are independent
Question number: 65
» Probability » Conditional Probability
Appeared in Year: 2009
Describe in Detail
The joint density of (X, Y) is
Find the conditional densities and E [X|Y = 1.5].
Explanation
First find the marginal distribution of X and Y
The conditional density of X|Y is
The conditional density of Y|X is
Question number: 66
» Probability » Standard Probability Distributions » Normal
Appeared in Year: 2011
Describe in Detail
Let the joint p. d. f. of (X, Y) be f (x, y) = e -y, 0 < x < y < ∞.
Obtain the probability P (X + Y ≤ 1).
Explanation
The Joint p. d. f. is
f (x, y) = e -y, 0 < x < y < ∞.
Let assume X + Y =U and Y = V, then X = U-V
Using Jacobian technique
The range is 0 < ∞, u ≤ v < ∞
The… (23 more words) …
Question number: 67
» Probability » Probability Generating Functions
Appeared in Year: 2009
Describe in Detail
Find the generating function of X whose probability density function is
P [X = r] =pq r-1, r = 1, 2, …, 0 < p < 1, q = 1 - p
Explanation
The probability generating function is defined as
Question number: 68
» Probability » Standard Probability Distributions » Exponential
Appeared in Year: 2009
Describe in Detail
Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless property.
Explanation
The property of memory less is that these distributions of “time from now to the next period” are exactly the same. The property is most easily explained in terms of “waiting times.
Suppose X is a discrete random variable whose values is a non-negative. In probability theory, a distribution is… (84 more words) …
Question number: 69
» Probability » Expectation
Appeared in Year: 2015
Describe in Detail
For random variables X, Y, show that
Explanation
We know that
Question number: 70
» Probability » Conditional Probability
Appeared in Year: 2015
Describe in Detail
Let X 1, X 2, …, X n be independent Poisson variates with E (X i) =µ i. Find the conditional distribution of
Explanation
For finding the conditional distribution, first find the distribution of sum of X 1, X 2, …, X n.
Assume that y-k = n
Similarly, for last two terms,
Given X 1, X 2, …, X n being… (67 more words) …
Question number: 71
» Probability » Laws of Total and Compound Probability
Appeared in Year: 2011
Describe in Detail
Verify the following identities:
(i)
(ii)
Explanation
Let A and B are two possible events in the sample space.
(i) Additive law of probability is
{Commutative property }
We know that
This shows that
(ii) This also show by additive law of probability
Let assume BUC = D, then
Putting the… (28 more words) …