Question 1

Suppose a random variable $$X$$ has the following probability mass function (pmf): $$p(x) = \frac{1}{5}$$ for $$x = 1,2,3,4,5. $$Which of the following statements is TRUE about this pmf?

Accepted Answer

The pmf is valid because $$\sum_{x=1}^5 p(x) = 1$$ and $$p(x) \geq 0$$ for all $$x.$$

Question 2

A fair coin is tossed 3 times. Let $$X be $$the number of heads. What is the probability that $$X = 2$$?

Accepted Answer

$$\frac{3}{8}$$

Question 3

Let $$X be a $$binomial random variable with parameters $$n=10$$ and $$p=0.7. $$What is the probability that $$X=7$$?

Accepted Answer

$$\binom{10}{7} (0.7)^7$ (0.3)^3$

Question 4

A hospital switchboard receives emergency calls at a rate of 2 every 30 minutes. What is the probability that at least 1 emergency call is received between 02:00 and 04:00 (a 2-hour period)?

Accepted Answer

$$1 - e$$^{-8}$$

Question 5

Suppose $$X is a $$geometric random variable with success probability $$p=0.2. $$What is the probability that the first success occurs on the 5th trial?

Accepted Answer

$$(0.8)^4$$ 	imes 0.2$$

Question 6

A drawer contains 5 brown socks and 3 green socks. Two socks are selected at random without replacement. Let $$X be $$the number of brown socks selected. What is $$P(X=1)$$?

Accepted Answer

$$\frac{15}{28}$$

Question 7

Let $$X be a $$continuous random variable with density function $$f(x) = 2x$$ for $$0 \u003c x \u003c 1. $$What is the cumulative distribution function (cdf) $$F(x)$$ for $$0 \u003c x \u003c 1$$?

Accepted Answer

$$F(x) = x^2$

Question 8

Suppose the time between arrivals of customers at a petrol station is exponentially distributed with mean 2 hours. What is the probability that no customers arrive for at least 3 hours?

Accepted Answer

Question 9

Suppose $$X is a $$Poisson random variable with mean $$\lambda = 2. $$What is the probability that $$X = 0$$?

Accepted Answer

$$e^{-2}$$

Question 10

A random variable $$X$$ has the following cumulative distribution function (cdf): $$F(x) = 0$$ for $$x \u003c 0$$, $$F(x) = x$$ for $$0 \leq x \u003c 1$$, $$F(x) = 1$$ for $$x \geq 1. $$What is the probability density function (pdf) of $$X$$?

Accepted Answer

$$f(x) = 1$$ for $$0 \leq x \u003c 1$$, 0 otherwise

Question 11

Suppose $$X is a $$continuous random variable with pdf $$f(x) = \frac{1}{	heta} e$$^{-x/	heta}$$ for x$$ \u003e 0$$ and $$	heta \u003e 0$$. What is the median of X$?

Accepted Answer

$$	heta \ln 2$$

Question 12

Let $$X be a $$standard normal random variable. What is $$P(-1 \u003c X \u003c 1)$$ approximately?

Accepted Answer

0.68

Question 13

Suppose $$X is a $$hypergeometric random variable with parameters $$N=20$$, $$K=7$$, $$n=5. $$What is the probability that $$X=3$$?

Accepted Answer

$$\frac{\binom{7}{3} \binom{13}{2}}{\binom{20}{5}}$$

Question 14

A random variable $$X$$ has the pdf $$f(x) = 3x^2$ for 0  0.5$?

Accepted Answer

Question 15

Suppose $$X is a $$continuous random variable with pdf $$f(x) = \frac{1}{2}$$ for $$0 \u003c x \u003c 2. $$What is the 75th percentile (the value $$q$$ such that $$P(X \leq q) = 0.75$$)?

Accepted Answer

$$1.5$$

Question 16

Let $$X be a $$random variable with pdf $$f(x) = \frac{1}{4}$$ for $$2 \u003c x \u003c 6. $$What is $$P(3 \u003c X \u003c 5)$$?

Accepted Answer

$$0.5$$

Question 17

Suppose $$X is a $$standard normal random variable and $$Y = 2X + 3. $$What is the distribution of $$Y$$?

Accepted Answer

$$Y \sim N(3, 4)$$

Question 18

Which of the following is a property of the exponential distribution?

Accepted Answer

It is memoryless: $$P(X \u003e s + t \mid X \u003e s) = P(X \u003e t)$$

Question 19

Suppose $$X is a $$random variable with pdf $$f(x) = 6x(1-x)$$ for $$0 \u003c x \u003c 1. $$What is the cdf $$F(x)$$ for $$0 \u003c x \u003c 1$$?

Accepted Answer

$$F(x) = 3x^2$ - 2x^3$

Question 20

Let $$U be a $$uniform random variable on $$[0,1]$$ and $$V = -\ln(U). $$What is the pdf of $$V$$?

Accepted Answer

$$f_V(v) = e$$^{-v}$$ for v$$ \u003e 0$$

Discrete and Continuous Random Variables: Probability Distributions and Functions

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