Whether the statement “The function whose graph is given at the right is a probability density function” is true or false.
Answer to Problem 1PT
The given statement is
Explanation of Solution
Definition used:
Probability density function:
(1) Every continuous random variable X has a probability density function f. That is,
(2) The probability density function f of a random variable X satisfies the condition
(3) The probabilities are measured on a scale from 0 to 1, it follows that
Formula used:
The slope of the tangent line is,
Calculation:
From the Figure, it is observed that the function
Therefore, the given graph is not continuous and it is not a probability density function.
This can be proved with the help of the formula (3) mentioned above.
Here,
Compute the equation of the line passing through the points
Compute the slope of the tangent line by substituting
Substitute
Therefore, the linear equation is,
Compute the value of b by substituting the point
Therefore, the linear function is
Compute the equation of the line passing through the points
Compute the slope of the tangent line by substituting
Substitute
Therefore, the linear equation is,
Compute the value of b by substituting the point
Therefore, the linear function is
Thus the function is
Compute
Further simplified as,
Hence, the given graph is not a probability density function by the definition mentioned above.
Therefore, the given statement is
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Chapter 8 Solutions
Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th
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