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The probability function of a continuous random variable is y = k ( x - 1), (1 ≤ x ≤ 2) then the value of the constant k is
- a)-1
- b)1
- c)2
- d)0
Correct answer is option 'C'. Can you explain this answer?
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The probability function of a continuous random variable is y = k ( x ...
The probability function of a continuous random variable must satisfy certain properties. One of these properties is that the integral of the probability function over its entire range must equal 1.
In this case, the probability function is given by y = k(x - 1), where k is a constant.
To find the constant k, we need to normalize the probability function by integrating it over its range and setting the result equal to 1.
∫[1,∞] k(x - 1) dx = 1
Using the integral properties, we can simplify the integral:
k ∫[1,∞] x - 1 dx = 1
k [ (1/2)x^2 - x ] [1,∞] = 1
Now we evaluate the integral:
k [ (1/2)(∞)^2 - ∞ ] - k [ (1/2)(1)^2 - 1 ] = 1
k (∞ - ∞) - k (1/2 - 1) = 1
Infinity minus infinity is undefined, so we can't evaluate the integral in this way. Therefore, the probability function y = k(x - 1) does not satisfy the property of being a valid probability function.
We need more information or a different formulation of the probability function to find the correct constant and ensure it satisfies the necessary properties.
In this case, the probability function is given by y = k(x - 1), where k is a constant.
To find the constant k, we need to normalize the probability function by integrating it over its range and setting the result equal to 1.
∫[1,∞] k(x - 1) dx = 1
Using the integral properties, we can simplify the integral:
k ∫[1,∞] x - 1 dx = 1
k [ (1/2)x^2 - x ] [1,∞] = 1
Now we evaluate the integral:
k [ (1/2)(∞)^2 - ∞ ] - k [ (1/2)(1)^2 - 1 ] = 1
k (∞ - ∞) - k (1/2 - 1) = 1
Infinity minus infinity is undefined, so we can't evaluate the integral in this way. Therefore, the probability function y = k(x - 1) does not satisfy the property of being a valid probability function.
We need more information or a different formulation of the probability function to find the correct constant and ensure it satisfies the necessary properties.
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