GATE EC 2014 Set 2 — Question 12
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Engineering Mathematics → Probability → Expectation & Variance
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Question
Let X be a random variable which is uniformly chosen from the set of positive odd numbers less than 100. The expectation, E[X], is __________
Correct answer
49.9 to 50.1
Solution
The random variable X is uniformly chosen from the set of positive odd numbers less than 100.The set of these numbers is .
This is an arithmetic progression.To find the number of terms, , in this set, we can use the formula for the -th term of an arithmetic progression: . Here, , , and the common difference .
So, there are 50 numbers in the set.Since X is uniformly chosen, the probability of choosing any particular number is .The expectation (or mean) of a discrete random variable is given by .
Since is constant, .We need to find the sum of the arithmetic series .
The sum of an arithmetic series is given by .
.Now, we can calculate the expectation:
.Alternatively, for a uniform distribution over a set of numbers in an arithmetic progression, the expectation is simply the average of the first and last terms.
.Therefore, the expectation E[X] is 50.
This is an arithmetic progression.To find the number of terms, , in this set, we can use the formula for the -th term of an arithmetic progression: . Here, , , and the common difference .
So, there are 50 numbers in the set.Since X is uniformly chosen, the probability of choosing any particular number is .The expectation (or mean) of a discrete random variable is given by .
Since is constant, .We need to find the sum of the arithmetic series .
The sum of an arithmetic series is given by .
.Now, we can calculate the expectation:
.Alternatively, for a uniform distribution over a set of numbers in an arithmetic progression, the expectation is simply the average of the first and last terms.
.Therefore, the expectation E[X] is 50.
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