Module 4: Random Variables and Introduction to Distributions

P(X ≤ x).

The cumulative distribution function (CDF) of a random variable X, typically denoted by F(x), is defined as ...

A: True

Q: All possible values of a random variable must be numbers. T/F?

A: Let X = the outcome of a die roll x can take on the values {1, 2, 3, 4, 5, 6}

Q: Consider the experiment of rolling a fair 6-sided die. Define a random variable for this experiment and list all possible outcomes.

A: True

Q: It is appropriate to plug values into a CDF to find probabilities in both the discrete and continuous cases. T/F?

a) Let X = the sum of rolling two six-sided dice Discrete S = {2,3,4,5,6,7,8,9,10,11,12} b) Let T = the lifetime of a light bulb Continuous S = { t | t 0 } c) Let H = the number of hurricanes that develop this year Discrete S = {0,1,2,3,4,5, ... } d) Let Z correspond to whether a mechanical part is defective or not Discrete S = {yes, no}

Q: Label each random variable as either discrete or continuous based on its description and briefly describe its sample space. a) Let X = the sum of rolling two six-sided dice b) Let T = the lifetime of a light bulb c) Let H = the number of hurricanes that d) Let Z correspond to whether a mechanical part is defective or not

1. f(x) ≥ 0 for all x ϵ R 2. f(x)dx=1 when integrating over x ϵ R

A probability density function must satisfy the following 2 rules:

1. 0 ≤ f(x) ≤ 1 for all x 2. Σ_(all x) f(x) = 1

A probability mass function must satisfy the following 2 rules:

f(x) = P(X = x)

A probability mass function of a discrete random variable X is defined as ...

function that associates a real number with each element in the sample space. Typically abbreviated with r.v. - In other words, a random variable is a function that simply assigns a number to each possible outcome in an experiment. - Mathematically, this says random variable X is X : S -> R - Almost always denoted with a capitol letter such as X, Y, and Z. An arbitrary observed value of a random variable is denoted with a lowercase letter.

A random variable is a ...

A discrete random variable that results in a dichotomy (only two possible outcomes) Typically denoted 0 (failure) and 1 (success)

Bernoulli Random Variable is ...

Determine the value of k that makes f(x)a valid pdf. ∫^(1)_(-1)k(3-x^(2))dx = k∫^(1)_(-1)3-x^(2)dx = k(3x-(1/3)x^(3)) |^(1)_(-1) k[3(1)-1(/3)(1)^(3)]-k[3(-1)-(1/3)(-1)^(3)]=k[3-(1/3)+3-(1/3)] k(16/3) = 1 k = 3/16

Ex: Consider (pictured) Determine the value of k that makes f(x) a valid pdf.

discrete random variables

F(x) = P(X ≤ x) = Σ_(t ≤ x)f(t) for -∞<x<∞ ( _________ random variables)

continuous random variables

F(x) = P(X ≤ x) = ∫^(x)_(∞)f(t) for -∞<x<∞ ( _________ random variables)

P(X ≤ 0.5) = P(-1 ≤ X ≤ 0.5) = ∫^(0.5)_(-1)(3/16)(3-x^(2))dx = ∫^(0.5)_(-1)(9/16)-(3/16)x^(2) dx = (9/16)x-(1/16)x^3 |^(0.5)_(-1) = 99/128 = 0.773

Find the probability that X ≤ 0.5.

a) Let M = the amount of milk produced yearly by a particular cow Continuous S={ m | m0 } b) Let P = the number of building permits issued by the City of Oxford this year Discrete S={0,1,2,3,4,5,...} c) Let F = the number of flips of a coin until 3 heads occur in succession Discrete S = {3,4,5,6,7,8, ...}

Label each random variable as either discrete or continuous based on its description and briefly describe its sample space. a) Let M = the amount of milk produced yearly by a particular cow b) Let P = the number of building permits issued by the City of Oxford this year c) Let F = the number of flips of a coin until 3 heads occur in succession

"the probability that r.v. X is equal to possible value x"

P(X = x) means ...

A: Let Y = the height of a randomly selected adult (in inches) S = { y | 48 y 84 } This means that y can take on values between 48 and 84, inclusive.

Q: Many random variables correspond to physical measurements. For instance, let Y be the height of a randomly selected adult (in inches). Discuss the sample space for the random variable Y. Note: The biologically plausible range for adult height is 48 to 84 inches.

A: False

Q: The possible values of a random variable are random. T/F?

A: d) Less than or equal to

Q: When you plug a value into a CDF, you will obtain the probability that the random variable is _____ that value. a) equal to b) greater than or equal to c) less than d) less than or equal to

by f(x).

Probability Density Function (pdf): For continuous random variables, the structure of the probabilities are determined by a probability density function (pdf), denoted ...

P(a ≤ X ≤ b)= ∫^(a)_(b) f(x)dx

Probability density functions are used to determine probabilities associated with continuous random variables by utilizing ...

discrete random variables. - They give _____all possible values_____ of the random variable X and the corresponding _____probability____ f(x)associated with each possible value. - They can be defined in _____numerous_____ ways. - The most common ways are via a table, piecewise function, or some other function notation.

Probability mass functions are used to determine probabilities associated with ...

A: False

Q: A probability mass function (pmf) can be used to describe the behavior of a continuous random variable.

∫^(y)_(0)1.5t^(2)+t dt = = 1.5(1/3)t^(3)+(1/2)y^(3)]^(0)_(y) = (1/2)y^(2)

The length of time required by students to complete a one-hour exam is a random variable with a probability density function given by (pictured). Determine the CDF for random variable Y.

P(X=a)=P(a ≤ X ≤ a) = ∫^(a)_(b) f(x)dx = 0

The probability that any continuous random variable is equal to any specific value is 0 utilizing ...

"the random variable X assumes the value x."

X = x means ...

Any random variable with a discrete random variable Coin flip (heads/tails) Defective / Operational Boy / Girl Yes / No True / False

examples of Bernoulli random variables?

No because all Σ_(all x) f(x) = 0.9 ≠ 1

is the following are valid probability mass functions? Explain.

Yes because 0 ≤ f(x) ≤ 1 for all x and Σ_(all x) f(x) = 0.3 + 0.1 +...+ 0.5 = 1

is the following are valid probability mass functions? Explain.

Yes because 0 ≤ f(x) ≤ 1 for all x and Σ_(all x) f(x) = Σ_(x=1) ^(6) 1/6 = 6(1/6) = 1

is the following are valid probability mass functions? Explain.

Yes, because: f(0) = p^(0) (1-p)^(1-(0)) = 1-p f(1) = p^(1) (1-p)^(1-(0)) = p(1-p) = p-p^(2) = p and Σ_(all x) f(x) = 1 - p + p = 1

is the following are valid probability mass functions? Explain.