GATE EE 2016 Set 2 — Question 43
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Engineering Mathematics → Probability → PMF, PDF & CDF
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Question
Let the probability density function of a random variable, X, be given as:
where u(x) is the unit step function.
Then the value of 'a' and Prob{X 0}, respectively, are
where u(x) is the unit step function.
Then the value of 'a' and Prob{X 0}, respectively, are
Correct answer
(A) 2, (1)/(2)
Solution
The given probability density function (PDF) is .
Here, is the unit step function, which is 1 for and 0 for . Similarly, is 1 for and 0 for . We can write the PDF piecewise:
(The value at is not uniquely defined but does not affect the integral).For to be a valid PDF, two conditions must be met:
.First, evaluate the integral for :
.Next, evaluate the integral for :
.Now, substitute these results back into the normalization equation:
.The first condition requires , which is satisfied by .Next, we need to find the probability . This is given by the integral of the PDF from to 0.
.
We have already calculated this integral:
.Substituting the value , we get:
.Thus, the value of 'a' is 2 and is . This corresponds to option (A).
Here, is the unit step function, which is 1 for and 0 for . Similarly, is 1 for and 0 for . We can write the PDF piecewise:
(The value at is not uniquely defined but does not affect the integral).For to be a valid PDF, two conditions must be met:
1. for all .
2..
Let's use the second condition to find the value of 'a'..First, evaluate the integral for :
.Next, evaluate the integral for :
.Now, substitute these results back into the normalization equation:
.The first condition requires , which is satisfied by .Next, we need to find the probability . This is given by the integral of the PDF from to 0.
.
We have already calculated this integral:
.Substituting the value , we get:
.Thus, the value of 'a' is 2 and is . This corresponds to option (A).
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