Question 1

Which of the following is a property of the cumulative distribution function (CDF) $$F_X(x)$$ for a continuous random variable $$X$$?

Accepted Answer

$$F_X(x) is a $$non-decreasing function

Question 2

Let $$X be a $$continuous random variable with probability density function (PDF) $$f_X(x). $$Which of the following statements is always true?

Accepted Answer

$$f_X(x) \geq 0$$ for all $$x$$

Question 3

Suppose the error in the reaction temperature $$X$$ has PDF $$f(x) = \frac{x^2$}{3}$$ for $$-1 \u003c x \u003c 2$$, and $$0$$ elsewhere. What is $$P(0 \u003c X \leq 1)$$?

Accepted Answer

$$\frac{1}{9}$$

Question 4

Given a continuous random variable $$X$$ with CDF $$F_X(x)$$, which of the following correctly expresses the probability that $$X$$ falls in the interval $$(a, b]$$?

Accepted Answer

$$F_X(b) - F_X(a)$$

Question 5

For a continuous random variable $$X$$, which of the following is true about the probability at a single point $$a$$?

Accepted Answer

$$P(X = a) = 0$$

Question 6

Suppose $$f(y) = \frac{5}{8b}$$ for $$\frac{2}{5}b \leq y \leq 2b$$ and $$0$$ elsewhere is the PDF for the winning bid $$y on a $$DOE project. What is the CDF $$F(y)$$ for $$\frac{2}{5}b \leq y \leq 2b$$?

Accepted Answer

$$F(y) = \frac{5}{8b}(y - \frac{2}{5}b)$$

Question 7

Given the CDF $$F(t)$$ for the maturity time $$T of a $$bond as $$F(t) = 0$$ for $$t \u003c 1$$, $$\frac{1}{4}$$ for $$1 \leq t \u003c 3$$, $$\frac{1}{2}$$ for $$3 \leq t \u003c 5$$, $$\frac{3}{4}$$ for $$5 \leq t \u003c 7$$, and $$1$$ for $$t \geq 7$$, what is $$P(T = 5)$$?

Accepted Answer

$$0$$

Question 8

For the same CDF as above, what is $$P(T \u003e 3)$$?

Accepted Answer

$$1 - F(3)$$

Question 9

Let $$f(x) = k\sqrt{x}$$ for $$0 \u003c x \u003c 1$$ and $$0$$ elsewhere. What value of $$k$$ makes $$f(x) a $$valid PDF?

Accepted Answer

$$k = \frac{2}{3}$$

Question 10

Given $$f(x) = k\sqrt{x}$$ for $$0 \u003c x \u003c 1$$ and $$k = \frac{2}{3}$$, what is the CDF $$F(x)$$ for $$0 \u003c x \u003c 1$$?

Accepted Answer

$$F(x) = \frac{2}{3}x$$^{3/2}$$

Question 11

Suppose $$X is a $$continuous random variable with PDF $$f_X(x). $$Which of the following integrals gives the probability that $$X$$ falls between $$a$$ and $$b$$?

Accepted Answer

$$\int_{a}^{b} f_X(x) dx$$

Question 12

If $$F_X(x) is $$the CDF of a continuous random variable $$X$$, what is the PDF $$f_X(x)$$?

Accepted Answer

$$f_X(x) = \frac{d}{dx} F_X(x)$$

Question 13

A continuous random variable $$X is $$defined over the interval $$[a, b]. $$Which of the following must be true for its PDF $$f_X(x)$$?

Accepted Answer

$$\int_{a}^{b} f_X(x) dx = 1$$

Question 14

Suppose the CDF of a random variable $$T is $$given as $$F(t) = 0$$ for $$t \u003c 1$$, $$\frac{1}{4}$$ for $$1 \leq t \u003c 3$$, $$\frac{1}{2}$$ for $$3 \leq t \u003c 5$$, $$\frac{3}{4}$$ for $$5 \leq t \u003c 7$$, and $$1$$ for $$t \geq 7. $$What is $$P(1.4 \u003c T \u003c 6)$$?

Accepted Answer

$$\frac{1}{2}$$

Question 15

For the same CDF as above, what is $$P(T \leq 5 \mid T \geq 2)$$?

Accepted Answer

$$\frac{F(5) - F(2)}{1 - F(2)}$$

Question 16

A medical device measures the time (in seconds) until a sensor detects a chemical. If the time $$T is $$modeled as a continuous random variable with PDF $$f_T(t)$$, which of the following is a valid interpretation of $$f_T(5)$$?

Accepted Answer

The relative likelihood that $$T is $$near 5

Question 17

Suppose $$X is a $$continuous random variable with PDF $$f_X(x) = \frac{x^2$}{3}$$ for $$-1 \u003c x \u003c 2. $$What is the value of the CDF $$F_X(2)$$?

Accepted Answer

$$1$$

Question 18

A continuous random variable $$X$$ has PDF $$f_X(x). $$Which of the following is NOT a property of $$f_X(x)$$?

Accepted Answer

$$f_X(x)$$ can be negative for some $$x$$

Continuous Random Variables, CDF, and PDF – Study Notes

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