1
GATE CSE 1999
MCQ (Single Correct Answer)
+1
-0.3
Suppose that the expectation of a random variable X is 5. Which of the following statements is true?
A
There is a sample point at which X has the value 5.
B
There is a sample point at which X has the value greater than 5.
C
There is a sample point at which X has a value greater than or equal to 5.
D
None of the above.
2
GATE CSE 1999
MCQ (Single Correct Answer)
+2
-0.6
Let X and Y be two exponentially distributed and independent random variables with mean $$\alpha $$ and $$\beta $$, respectively. If Z = min (X, Y), then the mean of Z is given by
A
$${1 \over {\alpha + \beta }}$$
B
$$\min \,(\alpha ,\,\beta )$$
C
$${{\alpha \,\beta } \over {\alpha + \beta }}$$
D
$${\alpha + \beta }$$
3
GATE CSE 1999
MCQ (Single Correct Answer)
+2
-0.6
Consider two events $${{E_1}}$$ and $${{E_2}}$$ such that probability of $${{E_1}}$$, Pr [$${{E_1}}$$] = 1/2, probability of $${{E_2}}$$, Pr[$${{E_2}}$$ = 1/3, and probability of $${{E_1}}$$ and $${{E_2}}$$, $$\left[ {{E_1}\,\,or\,\,{E_2}} \right]$$ = 1/5. Which of the following statements is /are true?
A
$$\Pr \,\left[ {{E_1}\,\,or\,\,{E_2}} \right]$$ is 2/3
B
Events $${{E_1}}$$ and $${{E_2}}$$ are independent
C
Events $${{E_1}}$$ and $${{E_2}}$$ are not independent
D
$$\Pr \,\left[ {{E_1}\,/\,{E_2}} \right] = 4/5$$
4
GATE CSE 1999
Subjective
+5
-0
Let $$\left( {\left\{ {p,\,q} \right\},\, * } \right)$$ be a semi group where $$p * p = q$$. Show that: (a) $$p * q = q * p,$$, and (b) $$q * q = q$$
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