Chapter 2 - Discrete Random Variables Theory

True or False: A discrete random variable has an associated probability density function (PDF) which gives the probability of each numerical value that the random variable can take.

False. A discrete random variable has an associated probability mass function (PMF) l which gives the probability of each numerical value that the random variable can take. Probability density function (PDF) relates to a continuous random variable.

True or False: An exact sequence for 5 tosses of a coin can be considered a Random Variable?

False. It is not considered a Random Variable because it does not have a specific numerical value.

True or False: A function of a random variable is NOT another random variable

False. The function of a random variable is a random variable.

Select the Geometric Random Variables: a. Number X tosses for heads to come up for the first time. b. Number X rolls of the die until a specific number is rolled for the first time. c. Number X games until someone wins the match.

a. Number X tosses for heads to come up for the first time. b. Number X rolls of the die until a specific number is rolled for the first time. c. Number X games until someone wins the match.

Select the terms that also mean Expected Value: a. 2nd moment b. E[X] c. Center of Gravity d. Mean

b. E[X] c. Center of Gravity d. Mean NOTE: The 2nd Moment refers to E[x^2]

Which is generally the quicker way to calculate Variance of a Discrete Random Variable? a. var[X] = E([X - E[x])^2] b. var[X] = E[X^2] - (E[X])^2 c. Both are the same.

b. var[X] = E[X^2] - (E[X])^2

Mean and Variance of a Linear Function of a Discrete Random Variable Let X be a random variable and let: Y = aX + b. where a and b are given scalars. Determine the Var[Y] in terms of X, a, and B

var[Y] = a^2 * var[X]

Select the choices that are valid discrete random variables: a) The sum of two rolls of the die b) The number of sixes in two rolls c) The second roll raised to the fifth power

All of the above. a) The sum of two rolls of the die b) The number of sixes in two rolls c) The second roll raised to the fifth power

What is a Poisson Random Variable a good approximation of?

Binomial Random Variable For large n and small p

Select the terms equivalent to the joint PMF pXY(x, y) a. P(X = x and Y = y) b. P({X = x} n {Y = y})

Both are equivalent. a. P(X = x and Y = y) b. P({X = x} n {Y = y})

What are the differences in parameters in Bernoulli problems vs Binomial problems?

Both have parameter, p, chance of success of a binary event. Binomial problems consider, n, number of fixed trials.

How do you calculate the marginal probability graphically?

Calculate the row sums and column sums. "Sum' it up" marginal (adjective) relating to or situated at the edge or margin of something.

Mean and Variance of a Linear Function of a Random Variable Let X be a random variable and let: Y = aX + b. where a and b are given scalars. Determine the E[Y] in terms of X, a, and B

E[Y] = aE[X] + b Using E[g(x)], where Y = g(x) = aX + b

How would one calculate the Joint PMF using the marginal probability and Conditional PMF

This is similar to the Total Probability Theorem.

True or False: There is a notion of independence of a random variable from an event or from another random variable.

True.