Exercises for Question Type 5: Finding the mode for a given PDF - IB

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Exercises for Question Type 5: Finding the mode for a given PDF - IB

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Question 1

For the piecewise PDF f (x) = ⎩ ⎨ ⎧ x^{2}, 1 - (x - 1)^{2}, 0, 0 < x < 1, 1 < x < 2, otherwise, derive the cumulative distribution function F (x) for all real x, and then identify the mode.

[6]

Question 2

Let f (x) = k x^{2} for 0 < x < 1 and f (x) = 0 otherwise.

(a)

Determine k to make f a PDF.

[3]

(b)

Find the mode of this distribution.

[2]

Question 3

Let f (x) = k (1 - (x - 1)^{2}) for 1 < x < 2 and f (x) = 0 otherwise.

(a)

Find the value of k .

[3]

(b)

Determine the mode of f .

[2]

Question 4

Find the mode of the probability density function f (x) = 3 x^{2} for 0 \leq x \leq 1 .

[3]

Question 5

Find the mode of the probability density function f (x) = \frac{3}{2} (1 - (x - 1)^{2}) for 1 \leq x \leq 2 .

[3]

Question 6

A random variable X has the piecewise PDF f (x) = ⎩ ⎨ ⎧ x^{2}, 1 - (x - 1)^{2}, 0, 0 < x < 1, 1 < x < 2, otherwise. Compute f^{'} (x) on each interval, determine the critical point, and hence find the mode.

[5]

Question 7

Suppose X has probability density function (PDF) f (x) = k x^{α} for 0 < x < 1, where α > 0 .

(a)

Determine the value of k .

[2]

(b)

Find the mode of X in terms of α .

[3]

(c)

Evaluate the mode for α = 2 .

[1]

Question 8

Let Y = ln X where X has PDF f_{X} (x) = 3 x^{2} for 0 < x < 1 and f_{X} (x) = 0 otherwise. Find the mode of Y .

[5]

Question 9

Consider the piecewise PDF f (x) = ⎩ ⎨ ⎧ x^{2}, 1 - (x - 1)^{2}, 0, 0 \leq x \leq 1, 1 < x \leq 2, otherwise.

Find the mode(s) of this distribution.

[4]

Question 10

For the PDF f (x) = ⎩ ⎨ ⎧ x^{2}, 1 - (x - 1)^{2}, 0, 0 \leq x \leq 1, 1 < x \leq 2, otherwise, determine the intervals on which f (x) is increasing and decreasing, and use this to justify the mode location.

[4]

Question 11

Given the probability density function (PDF): f (x) = ⎩ ⎨ ⎧ x^{2}, 1 - (x - 1)^{2}, 0, 0 < x \leq 1 1 < x < 2 otherwise

(a)

State the mode of the distribution.

[1]

(b)

Find f^{''} (x) for the interval 1 < x < 2 .

[2]

(c)

Justify that the mode is a maximum.

[2]

Question 12

Let Y = X^{3} where X has PDF f_{X} (x) = 3 x^{2} for 0 < x < 1 and 0 otherwise. Find the mode of Y .

[3]