Generate the population in the next iteration by using Roulette-Wheel criterion:

 Consider a dataset of five observations given in the following table, each of which has two features f₁ and f₂:

Assume the number of cluster  and the real number . Also, assume the initial cluster centers as  and . Apply fuzzy c-mean algorithm to find the modified cluster center after one iteration.

Approximate the function  for , by solving 1-2-1 network, using Back propagation algorithm. The weighted structure and initial input are as follows:
Weighted structures are:
 and bias 
 and bias 
The initial input is 1.
Draw the architecture of the model. Perform one iteration.

 Out of three genetic operators viz. selection, cross-over and mutation, list and justify which operator or combination there of will be required for the following:

i) To fill the population with copies of the best individual from the population.
ii) For the convergence of an algorithm to good but sub-optimal solution.

Implement AND function using McCulloch-Pitts neuron.

 Let A and B be two Fuzzy sets as given below:


Determine the following:
i) Universe of discourse for Set A and Set B.
ii) Complements of Set A and Set B
iii) 
iv) 

 Consider the fuzzy sets A and B defined on the interval [0, 5]. Their membership functions are:

$

and 

Determine the membership function and graph them for each of the following:

i) A^C, B^C

ii) 

iii) 

iv) 

v) 

Find the length and order of the following schema:

i) 

ii) 

iii) 

iv) 

 Consider the ADALINE filter with three neurons in the input layer having weights 3,1 and − 2and the input sequence ......., ,0,0,0 − 0,0,0,5,4 ,.....}. Find the filter output.

 Consider the single layer perception given below:

The activation function is:

Obtain the output for each of the following input pattern:

 A single layer neural network is to have six inputs and three outputs. The outputs are continuous over the range 0 to 1. Now answer the following:
i) How many neurons are required?
ii) What are the dimensions of the weight matrix?
iii) What kind of transfer function could be used?
iv) Is a bias required? Give reasons.

Verify whether the Genetic Algorithm (GA) improves the solution from one generation to the next generation, for the function given below:
Maximize:

Subject to:

Assume that chromosomes of length 6 are created at random and modified by Roulette-Wheel selection.

Solve the network to approximate the function:

for , choosing the initial weights and bias as the random numbers.

 Determine the fuzzy relation T as a composition between the fuzzy relations R and S given below by using max-min and max-product:

 State whether the following statements are True or False. Give short proof or a counter example in support of your answer.
a) In the Hopfield network, the neurons belonging to the same layer receive input from the neurons of the previous layer and send their value only to the neurons of the next layer.
b) The length of chromosomes to determine the maximum value of the set:

 is 12.
c) The fuzzy relation (R) given below, is an equivalence relation.

d) The Self Organizing Map (SOM) is a supervised learning technique.
e) In a single layer neural network, if , then the output is -1.

 Determine the fuzzy relation T as a composition between the fuzzy relations R and S given below by using max-min and max-product:

State whether the following statements are True or False. Give short proof or a counter example in support of your answer.
a) In the Hopfield network, the neurons belonging to the same layer receive input from the neurons of the previous layer and send their value only to the neurons of the next layer.
b) The length of chromosomes to determine the maximum value of the set:

 is 12.
c) The fuzzy relation (R) given below, is an equivalence relation.

d) The Self Organizing Map (SOM) is a supervised learning technique.
e) In a single layer neural network, if , then the output is -1.

 एक मुहल्ले में रहने वाले वैवाहिक जोड़ों में, पति के एक स्कूल बोर्ड चुनाव में वोट देने की प्रायिकता 0.21 है, पत्नी के चुनाव में वोट देने की प्रायिकता 0.28 है और दोनों के वोट देने की प्रायिकता 0.15 है। इस बात की प्रायिकता क्या होगी कि 
i) उनमें से कम से कम एक वोट देगा?
ii) उनमें से कोई भी वोट नहीं देगा?

मान लीजिए X प्राचल  और , वाला एक गामा चर है जिसके  और  हैं।  और  ज्ञात कीजिए। गामा चर का आघूर्ण जनक फलन भी ज्ञात कीजिए और इस प्रकार सिद्ध कीजिए कि आघूर्ण जनक फलन से X का माध्य 6 और X का प्रसरण 3 है। 

 20 पदों का माध्य और मानक विचलन क्रमशः 10 और 2 पाया गया है। जाँच करते समय यह पाया गया कि मान 8 वाला एक पद गलत है। यदि गलत पद को हटा दिया जाए तो माध्य और मानक विचलन परिकलित कीजिए।