Probability (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 64 - 71 of 72

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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.

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The joint density of (X, Y) is

Find the conditional densities and E [X|Y = 1·5].

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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).

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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

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Explain “Memoryless property” of a distribution. Show that the exponential distribution has memoryless property.

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For random variables X, Y, show that

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Let X ₁, X ₂, …, X _n be independent Poisson variates with E (X _i) =µ _i. Find the conditional distribution of

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Verify the following identities:

(i)

(ii)