Q: Suppose $$X is a $$discrete random variable with PMF $$p_X(0)=0.2$$, $$p_X(1)=0.3$$, $$p_X(2)=0.4$$, $$p_X(3)=0.1. $$What is $$E[X]$$?

$$0 \times 0.2 + 1 \times 0.3 + 2 \times 0.4 + 3 \times 0.1 = 1.3$$

Q: A life insurance company offers a $$1,000 policy for a premium of 14$ to a 50-year-old male. The probability of death in a year is 0.006$. What is the expected gain for the company?

$$E[X] = 14 \times 0.994 + (-986) \times 0.006$$

Question 1

Which of the following statements best describes a discrete random variable?

Accepted Answer

It takes values from a countable set, such as integers.

Question 2

Suppose you flip two fair coins. Let $$X be $$the number of heads. What is the probability mass function (PMF) of $$X$$?

Accepted Answer

$$p_X(x) = \frac{1}{4}$$ for $$x=0$$, $$\frac{1}{2}$$ for $$x=1$$, $$\frac{1}{4}$$ for $$x=2$$

Question 3

Which of the following is NOT a property of a valid probability mass function (PMF) $$p(x)$$?

Accepted Answer

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

Question 4

A box contains three balls labeled 1, 2, and 2. If $$X is $$the number written on the ball drawn at random, what is $$P(X=2)$$?

Accepted Answer

$$\frac{2}{3}$$

Question 5

Given a discrete random variable $$X$$ with PMF $$p_X(x)$$, the cumulative distribution function (CDF) $$F_X(x) is $$defined as:

Accepted Answer

$$F_X(x) = \sum_{z \leq x} p_X(z)$$

Question 6

Suppose $$X is a $$discrete random variable with PMF $$p_X(0)=0.2$$, $$p_X(1)=0.3$$, $$p_X(2)=0.4$$, $$p_X(3)=0.1. $$What is $$E[X]$$?

Accepted Answer

$$0 	imes 0.2 + 1 	imes 0.3 + 2 	imes 0.4 + 3 	imes 0.1 = 1.3$$

Question 7

A life insurance company offers a $$1,000 policy for a premium of 14$ to a 50-year-old male. The probability of death in a year is 0.006$. What is the expected gain for the company?

Accepted Answer

$$E[X] = 14 	imes 0.994 + (-986) 	imes 0.006$$

Question 8

Which formula correctly expresses the variance of a discrete random variable $$X$$?

Accepted Answer

$$Var(X) = E[X^2$] - (E[X])^2$

Question 9

If $$X$$ and $$Y$$ are independent random variables, which of the following is true?

Accepted Answer

$$Var(X+Y) = Var(X) + Var(Y)$$

Question 10

Let $$X be a $$discrete random variable with $$E[X]=2$$ and $$Var(X)=3. $$What is $$Var(5X-4)$$?

Accepted Answer

$$Var(5X-4) = 25Var(X)$$

Question 11

Suppose $$X$$ takes values in $$\{2,3,4\}$$ with $$P(X=x) = c(x-1)$$ for $$x=2,3,4. $$What is the value of $$c$$?

Accepted Answer

$$c=\frac{1}{3}$$

Question 12

A company tests light bulbs and records their lifetimes (in years) as $$X$$ with PMF $$p(1)=0.1$$, $$p(2)=0.2$$, $$p(3)=0.3$$, $$p(4)=0.25$$, $$p(5)=0.15. $$What is the conditional probability $$P(X=5|X\u003e3)$$?

Accepted Answer

$$\frac{0.15}{0.25+0.15} = \frac{0.15}{0.4} = 0.375$$

Question 13

Given a discrete random variable $$X$$ with PMF $$p_X(x)$$, which of the following is the correct formula for the expected value of a function $$f(X)$$?

Accepted Answer

$$E[f(X)] = \sum_x f(x) p_X(x)$$

Question 14

Suppose $$X is $$the number of flips of a fair coin until the first head appears. What is the PMF of $$X$$?

Accepted Answer

$$p_X(x) = \frac{1}{2^x}$$ for $$x=1,2,3,...$$

Question 15

If $$X is a $$discrete random variable with PMF $$p_X(x)$$, which statement is true about the CDF $$F_X(x)$$?

Accepted Answer

It is a step function that jumps at each possible value of $$X.$$

Question 16

Suppose $$X is a $$discrete random variable with $$E[X]=3$$ and $$Var(X)=2. $$What is the standard deviation of $$X$$?

Accepted Answer

$$\sqrt{2}$$

Question 17

Which of the following best describes the difference between a PMF and a CDF for a discrete random variable?

Accepted Answer

PMF gives the probability at each value; CDF gives the probability at or below a value.

Question 18

If $$X is a $$discrete random variable and $$a$$ and $$b$$ are constants, which of the following is true?

Accepted Answer

$$E[aX+b] = aE[X] + b$$

L4.2 Discrete Random Variables and Probability Distributions: STA103 Lecture 4 Study Notes

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