Let E1, E2, E3 and E4 be arbitrary events. Write the following events in set notations:
i) not more than one of E1, E2, E3, E4.
ii) one and only one of E1, E2, E3, E4.
iii) E1 and at least one of E2, E3, E4.
iv) none of E2, E3 and E4 using E1.
b) Let the probability density function of r.v. X be
and if and
, find Cov(u,v). Also check the independence of u and v.
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___ ______ ________ ________ _________ ______ _____ __________ ___ _____.
_________ _____ _______ __________ ______ _______ _________ _______ __________ __________ __________ ________.
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मान लीजिए कि r.v. X का प्रायिकता घनत्व फलन निम्नलिखित है :
और यदि और
हो, तो Cov(u,v) ज्ञात कीजिए। u और v के स्वांतन्त्र्य की भी जाँच कीजिए।
मानक विचलन 5 वाले मध्यककुदी बंटन के लिए चतुर्थ केंद्रीय आघूर्ण m4 ज्ञात कीजिए।
6 observations on $(X, Y)$ yielded the following data:
\sum X_i = 30, \sum Y_i = 180, \sum X_i Y_i = 1000,
\sum X_i^2 = 200, \sum Y_i^2 = 5642.
i) Determine the correlation coefficient between X and Y.
ii) Given X = 10, what will be the predicted value of Y?
iii) Given Y = 15, what will be the predicted value of X?
Let X1 be an observation from an exponential distribution with the p.d.f.
$
Test the null hypothesis that the mean of the distribution is against the alternative hypothesis that is
. The null hypothesis is accepted if and only if the observed value of the random variable is less than 3. Find the probabilities of type-I and type-II errors.
b) The mean and standard deviation of 20 items is found to be 10 and 2 respectively. At the time of checking it was found that one item having value 8 was incorrect. Calculate the mean and standard deviation if the wrong item is omitted.
he mean I.Q. of a large number of children of age 14 was 100 and standard deviation 16. Assuming that the distribution was normal, find
i) the percentage of children having I.Q. under 80.
ii) the limits in which the I.Q. of the middle 40% of the children will lie.
You may like to use the following values:
Solve the following LPP graphically:
aximize:
$
subject to the constraints:$
$
$
b) Find all values of k for which the vectors:$
are linearly independent.
किसी पुल को पार करते समय एक कार के टायर सपाट होने की प्रायिकता 0.00005 है। पुल पर करने वाली 10,000 कारों के लिए वह प्रायिकता ज्ञात कीजिए जबकि
i) ठीक 2 कारों के टायर सपाट होंगे।
ii) ज्यादा से ज्यादा 2 कारों के टायर सपाट होंगे।