Q: If $$X is $$exponentially distributed with mean $$\theta$$, what is the probability density function $$f(x)$$ for $$x \geq 0$$?

$$f(x) = \frac{1}{\theta} e$$^{-x/\theta}$$

Question 1

Suppose the time between arrivals at a drive-in teller machine, $$X$$, is uniformly distributed between 5 and 15 minutes. What is the value of the constant $$c in $$the probability density function $$f(x)$$?

Accepted Answer

$$c = 0.1$$

Question 2

Given the uniform distribution for $$X ($$time between arrivals at a drive-in teller machine) on the interval $$[5, 15]$$, what is $$P(X \leq 10)$$?

Accepted Answer

$$0.5$$

Question 3

For the same uniform distribution, what is $$P(X \u003c 20)$$?

Accepted Answer

$$1$$

Question 4

For a continuous random variable $$X$$ with probability density function $$f(x)$$, which of the following must be true for $$f(x) to be a $$valid probability distribution?

Accepted Answer

$$\int_{-\infty}^{\infty} f(x) dx = 1$$

Question 5

Suppose a production run generated 1000 items: 150 with 1 defect, 600 with 2 defects, and 250 with 3 defects. What is the expected number of defects per item?

Accepted Answer

$$2.1$$

Question 6

Given a discrete probability distribution with $$p(1) = 0.3$$, $$p(2) = 0.3$$, $$p(3) = 0.4$$, what is the variance $$\$$sigma^2$$ if the possible values are 1$$, 2, 3$$?

Accepted Answer

$$0.49$$

Question 7

For a normal distribution with mean $$\mu = 1000$$ and standard deviation $$\sigma = 50$$, what is the $$z-$$score for $$X = 900$$?

Accepted Answer

$$z = -2$$

Question 8

If $$Z is a $$standard normal variable, what is $$P(Z \leq 1)$$?

Accepted Answer

$$0.8413$$

Question 9

A company’s monthly rate of return is given by $$R_t = \frac{(P_t - P_{t-1}) + D_t}{P_{t-1}}. If$$ $$P_t = 110$$, $$P_{t-1} = 100$$, and $$D_t = 0$$, what is the monthly rate of return?

Accepted Answer

$$0.10$$

Question 10

For a normal distribution $$N(1000, 2500)$$, what is the probability that $$X \leq 900$$?

Accepted Answer

$$0.0228$$

Question 11

For a uniform distribution on $$[a, b]$$, what is the formula for the mean $$\mu$$?

Accepted Answer

$$\mu = \frac{a + b}{2}$$

Question 12

For a continuous random variable $$X$$ with mean $$\mu$$ and variance $$\$$sigma^2$$, which formula gives the variance?

Accepted Answer

$$\$$sigma^2$$ = E[(X - \mu)^2$] = \int_{-\infty}^{\infty} (x - \mu)^2$ f(x) dx$$

Question 13

If $$X is $$exponentially distributed with mean $$	heta$$, what is the probability density function $$f(x)$$ for $$x \geq 0$$?

Accepted Answer

$$f(x) = \frac{1}{	heta} e$$^{-x/	heta}$$

Question 14

Given a normal distribution with mean $$\mu$$ and standard deviation $$\sigma$$, what is the $$z-$$score transformation for a value $$x$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma}$$

Question 15

If $$X is a $$continuous random variable with PDF $$f(x)$$, what is the probability that $$X$$ falls between $$a$$ and $$b$$?

Accepted Answer

$$P(a \leq X \leq b) = \int_a^b f(x) dx$$

Question 16

For a discrete random variable $$X$$ with probability mass function $$p(x)$$, what is the formula for the expected value $$E(X)$$?

Accepted Answer

$$E(X) = \sum_x x p(x)$$

Question 17

If $$Z is a $$standard normal variable, what is $$P(Z \geq 2)$$?

Accepted Answer

$$0.0228$$

Question 18

A process produces items with the following probabilities for defects: $$p(1) = 0.3$$, $$p(2) = 0.3$$, $$p(3) = 0.4. $$What is the probability that a randomly selected item has more than 1 defect?

Accepted Answer

$$0.7$$

Question 19

For a normal distribution, which of the following statements is true?

Accepted Answer

The mean, median, and mode are all equal.

Probability Distributions: Discrete and Continuous Random Variables

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