Probability (ISS Statistics Paper I (Old Subjective Pattern)): Questions 68 - 72 of 72

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Question number: 68

» Probability » Standard Probability Distributions » Exponential

Appeared in Year: 2009

Essay Question▾

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… (218 more words) …

Question number: 69

» Probability » Expectation

Appeared in Year: 2015

Essay Question▾

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For random variables X, Y, show that

V[Y]=EX[V(Y X)]+VX[E(Y X)]

Explanation

We know that

V[Y]=E(Y2)[E(Y)]2

=EX[E(Y2)X][E(Y)]2

=EX[V(YX)+… (104 more words) …

Question number: 70

» Probability » Conditional Probability

Appeared in Year: 2015

Essay Question▾

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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

X1,X2,,Xn i=1nXi=y.

Explanation

For finding the conditional distribution, first find the distribution of sum of X 1, X 2, …, X n.

P(i=1nXi=y)=P(X1+X2++Xn=y)… (732 more words) …

Question number: 71

» Probability » Laws of Total and Compound Probability

Appeared in Year: 2011

Essay Question▾

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Verify the following identities:

(i) P(AB)=P(A)+P(BAc)

(ii) P(ABC)=P(A)+P(B)+(C)P(AB)P(BC)P(AC)+P(ABC)

Explanation

Let A and B are two possible events in the sample space.

(i) Additive law of probability is

P(AB)=P(A)+P(B)P(AB)

=P(A)+P(B… (339 more words) …

Question number: 72

» Probability » Distribution Function » Standard Probability Distributions

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Show that f(x)=ax3,a<x,a>0 is a probability density function for an appropriate value of a. Upto what order do the moments of this p. d. f. exist?

Explanation

To find the value of a, we known that the probability density function under the range of x is equal to one.

af(x)dx=1

aa1x3dx=1

a[12x… (114 more words) …

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