GATE EE 2017 Set 2 — Question 40
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Engineering Mathematics → Probability → Expectation & Variance
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Question
A person decides to toss a fair coin repeatedly until he gets a head. He will make at most 3 tosses. Let the random variable denote the number of heads. The value of , where denotes the variance, equals
Correct answer
(C) (7)/(64)
Solution
Let denote a head and denote a tail. The coin is fair, so .
The person tosses the coin repeatedly until a head appears, making at most 3 tosses.
The possible outcomes and their probabilities are:
Check: . The distribution is valid.Next, calculate the expected value :
Next, calculate the expected value of , :
Finally, calculate the variance using the formula :
To subtract, find a common denominator:
The value of is .The final answer is
The person tosses the coin repeatedly until a head appears, making at most 3 tosses.
The possible outcomes and their probabilities are:
1.H: Head on the first toss.
. Number of heads .2.TH: Tail on the first, Head on the second.
. Number of heads .3.TTH: Tail on the first, Tail on the second, Head on the third.
. Number of heads .4.TTT: Tail on the first, Tail on the second, Tail on the third (no head in 3 tosses).
. Number of heads .Now, let's find the probability distribution of :Check: . The distribution is valid.Next, calculate the expected value :
Next, calculate the expected value of , :
Finally, calculate the variance using the formula :
To subtract, find a common denominator:
The value of is .The final answer is
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