ENVE 3510: Concepts 1

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If the jointly continuous random variables X and Y have the density function f(x,y) = (2/3)(2x + y) for 0 ≤ y ≤ 1 (and 0 elsewhere), then px(x) = _____ for 0 ≤ x ≤ 1.

(1/3)(4x + 1)

If the jointly continuous random variables X and Y have the density function f(x,y) = (2/3)(2x + y) for 0 ≤ x ≤ 1 (and 0 elsewhere), then py(y) = _____ for 0 ≤ y ≤ 1.

(2/3)(1 + y)

The population correlation between two jointly distributed random variables is always between _____.

-1 and 1

For random variables X and Y, their correlation ρX,Y is in which range?

-1 ≤ ρX,Y ≤ 1

A pop quiz consists of three true-false questions (with two possible answers each) and one multiple-choice question with five possible answers. The four questions are independent. How many different sets of answers are there?

If independent random variables X and Y have standard deviations σX = 3 and σY = 4, then the standard deviation of X + Y is σX+Y = ______.

If random variables X and Y are independent, then which of the following must be true?

Cov(X,Y) = 0 ρX,Y = 0

The of events A and B is the set of outcomes that belong both to A and to B.

Intersection

Consider the table giving the joint and marginal probability mass functions for the diameter X (in mm) and volume Y (in cc) of some candy canes. Suppose X and Y are independent. Complete the table that describes pY|X(y|6) by choosing the correct values of a and b.

a = 0.30 and b = 0.70

Consider the table giving the joint and marginal probability mass functions for the diameter X (in mm) and volume Y (in cc) of some candy canes.Suppose X and Y are independent. Complete the table that describes pY|X(y|6) by choosing the correct values of a and b.

a = 0.30 and b = 0.70

If X1,..., Xn are random variables and c1,..., cn are constants, then the random variable c1X1 + ... + cnXn has mean ______.

c1μX1 + ... + cnμXn

Let X and Y be jointly distributed random variables with standard deviations σX and σY. The of X and Y is ρX,Y = Cov(X,Y)σXσY

correlation

Let X and Y be random variables with means μX and μY. The of X and Y is Cov(X,Y) = μ(X−μX)(Y−μY).

covariance

The _____ distribution function of a continuous random variable X is F(x) = P(X ≤ x) = ∫x−∞f(t) dt.

cumulative

If X1,..., Xn are random variables and c1,..., cn are constants, then the random variable c1X1 + ... + cnXn is called a combination of X1,..., Xn.

linear

The of a continuous random variable X is the point xm such that P(X ≤ xm) = 0.5.

median

Match each probability mass function on the left with its definition on the right.

p(x,y) = P(X=xandY=y) pX(x) = ∑yp(x,y) pY(y) = ∑xp(x,y)

For independent jointly discrete or jointly continuous random variables X and Y, match each expression on the left with an equivalent expression on the right. (Assume that none of the expressions is zero.)

pY|X(y|x)=pY(y) fY|X(y|x)=fY(y) p(x,y)=pX(x)pY(y) f(x,y)=fX(x)fY(y)

What is the value of the correlation between two independent random variables?

A mountaineer attaches a backpack to two long pieces of nylon webbing of the same length, each wound on its own spool. Each spool contains a taped webbing joint having essentially zero strength with probability 0.1. One spool having a joint is independent of the other spool having a joint. The spools are unwound to lower the backpack off a tall cliff. What is the probability the backpack falls because both spools have a taped joint?

0.01

The probabilities of an arborist breaking a saw chain in a day's work are 0.004 for a small climbing saw, 0.008 for a medium limbing saw, and 0.04 for a large felling and bucking saw. What is the probability that he will break a chain on a day in which he spends 1/2 of his time climbing, 3/8 limbing, and 1/8 felling and bucking?

0.01

The variance of a random variable X with probability density function f(x) = 2x for 0 ≤ x ≤ 1 (and f(x) = 0 otherwise) is _____.

1/18

Let X be the result of rolling a loaded six-sided die with probabilities as shown:P(X = x) 1/12,1/6,1/6,1/6,1/6,1/4 The population variance of X is approximately _____.

2.74

If the jointly continuous random variables X and Y have the density function f(x,y) = (2/3)(2x + y) over the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (and 0 elsewhere), then P(0.5 ≤ x ≤ 1 and 0 ≤ y ≤ 1) = ______.

2/3

The mean of a random variable X with probability density function f(x) = 2x for 0 ≤ x ≤ 1 (and f(x) = 0 otherwise) is μx = _____.

2/3

In today's weather forecast, the probability of rain is 1/2, the probability of lightning is 3/10, and the probability of lightning given rain is 2/5. Find the probability of rain given lightning.

2/5

Suppose X, Y, and Z are independent random variables with variances σ2X = 3, σ2Y = 4, and σ2Z = 5. Find the variance of W = X + 2Y +Z.

Which of these random variables is a linear combination of random variables X, Y, and Z?

2X + Y - Z 2X + 4Y - 3Z

Suppose pencil lengths, in inches, are randomly distributed with variance 0.6 square inches. There are 2.54 centimeters in 1 inch. What is the variance of the pencil lengths in square centimeters? (Round to two places after the decimal point.)

3.87

Let X be the result of rolling a loaded six-sided die with probabilities as shown:P(X = x) 1/12,1/6,1/6,1/6,1/6,1/4 The mean value of X is approximately _____.

3.92

For a random variable X with mean 5 and variance 4, the variance of X + 3 is _____.

A deck of 52 playing cards consists of 4 suits (Clubs, Diamonds, Hearts, Spades) of 13 cards each (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). A randomly chosen card is drawn. The probability it is an Ace or Club is _____. (Answer with a reduced fraction.)

4/13

For jointly distributed random variables X and Y with standard deviations σX and σY, the correlation of X and Y is ρX,Y = ______.

Cov(X,Y)/σXσY

Which of the following is the conditional expectation of Y given X for jointly continuous random variables X and Y?

E(YX = x) = ∫∞−∞yfYX(yx) dy

Which of the following sentences describe a random variable?

Let H be the height, in inches, of a randomly chosen tree in a park. Let D be the number of defective bulbs in a box. Let K be the number of knots in a two-by-four board randomly chosen from a lumber warehouse.

Suppose a simple random sample of size 9 is drawn from a population with mean 180 and variance 81. Match each property of the sample mean on the left with its value one right.

Mean of X = 180 Variance of X = 9 Standard deviation of X = 3

Which of the following expressions represent the proportion of times that event A would occur if the experiment were repeated over and over? (Select all that apply.)

P(A) The probability of A

The probability mass function of a discrete random variable X is p(x) = _____.

P(X = x)

Which probability statement about all sets of numbers S and T is true for independent random variables X and Y?

P(X ∊ S ∩ Y ∊ T) = P(X ∊ S)P(Y ∊ T)

If X1, ..., Xn are random variables and c1, ..., cn are constants, then the random variable c1X1 + ... + cnXn has variance _____.

c21σ2x1 + ... + c2nσ2xn + 2 ∑n − 1j = 1∑nj = j + 1c1cjCov(XiXj)

's Inequality says that if X is a random variable with mean μX and standard deviation σX, then P(|X - μX| ≥ kσX) ≤

chebyshev

A _____ is a distinct group of objects that can be selected without regard to order.

combination

The of an event A is the set of outcomes in the sample space that do not belong to A.

complement

A quarterback, when throwing the football, has one of the following mutually exclusive outcomes (with associated probabilities): complete pass (1/2), incomplete pass (1/4), interception (1/8), or fumble (1/8). Match each event to is probability.

complete pass or incomplete pass, 3/4 complete pass or fumble, 5/8 incomplete pass or interception, 3/8

The probability density function of Y given X for jointly continuous random variables X and Y is fY|X(y|x) = f(x,y)fX(x).

conditional

The probability mass function of Y given X for jointly discrete random variables X and Y is pY|X(y|x) = p(x,y)pX(x).

conditional

Probabilities of a random variable are given by areas under a curve called its probability density function.

continuous

Let X be the number of heads in ten tosses of a fair coin. X is a _____ random variable.

discrete

The _____ of a continuous random variable X is ∫∞−∞x f(x) dx. (Choose all correct answers.)

expectation expected value mean

The _____ of a continuous random variable X is ∫∞−∞x f(x) dx. (Choose all correct answers.) Multiple select question.

expectation mean expected value

The conditional probability density function of Y given X for jointly continuous random variables X and Y is fY|X(y|x) = ______.

f(x,y)/fX(x)

True or false: Events A and B are mutually exclusive if their union is nonempty.

false

Random variables X and Y are if ρX,Y = 0.

independent

The jointly continuous random variables X and Y are if f(x,y) = fX(x)fY(y).

independent

The jointly discrete random variables X and Y are if p(x,y) = pX(x)pY(y).

independent

Two random variables X and Y are if, for all sets of numbers S and T, P(X ∊ S and Y ∊ T) = P(X ∊ S)P(Y ∊ T).

independent

Let X be the number of heads in a fair coin flip (0 or 1). Let Y be the number rolled on a fair six-sided die (1, 2, 3, 4, 5, or 6). Match each expression on the left to its value on the right.

p(1, 5)=112 pX(1)=12 pY(5)=16

The median of a of a continuous random variable X is _____.

the point xm such that P(X ≤ xm) = 0.5 the point xm such that F(xm) = 0.5

The variance of the sum of a random variable and a constant is _____.

the variance of the random variable

The events A and B are mutually exclusive if _____.

they have no outcomes in common their intersection is the empty set

True or false: The mean of a sum of random variables is the sum of their means.

true

True or false: The probability mass function p(x) of a discrete random variable X is p(x) = P(X = x).

true

For random variables X and Y, if Cov(X,Y) = 0, then X and Y are said to be _____.

uncorrelated

For the random variable X with probability density function f(x) = 2x for 0 ≤ x ≤ 1 (and f(x) = 0 otherwise), the cumulative distribution function F(x), over the interval 0 ≤ x ≤ 1, is F(x) = _____.

x^2

Consider tossing two coins. The sample space is S = {HH, HT, TH, TT}. Let event A = {HH} (both heads). Let event B = {HH, HT, TH} (at least one head). Which of these sets is A ∩ B?

{HH}

For jointly discrete random variables X and Y, match the expectation (or mean) on the left with its equivalent on the right.

μX=∑xxpX(x) μXY = y,=∑xxpXY(xy) μYX = x,=∑yypYX(yx)

Select all that apply Which two of these are equal to Cov(X,Y)?

μXY - μXμY μ(X-μX)(Y-μY)

Which two of these are equal to Cov(X,Y)?

μXY - μXμY μ(X-μX)(Y-μY)

For independent random variables X and Y, the variance of X + Y is σ2X+Y = ______.

σ2X+σ2Y

The of a continuous random variable X is ∫∞−∞(x − μx)2 f(x) dx.

variance

Consider tossing two coins. The sample space is S = {HH, HT, TH, TT}. Let event A = {HH} (both heads). Which of these sets is AC?

{HT, TH, TT}

If X is a random variable and a is a constant, the standard deviation of aX is σaX = _____

|a|σX

For a random variable X and a constant b, the mean of X + b is _____.

μX + b

The output of a rope splicing shop includes cosmetic defects in 1/10 of its single braid splices, 1/8 of its double braid splices, and 1/20 of its 16-strand splices. For a large order of splices, 1/3 are in single braid rope, 1/2 are in double braid rope, and 1/6 are in 16-strand rope. What is the probability of a cosmetic defect in a randomly-chosen splice from this order?

0.104

Here are mutually exclusive outcomes of a particular quarterback throwing a pass, with associated probabilities: complete pass (1/2), incomplete pass (1/4), interception (1/8), or fumble (1/8). (Treat these outcomes as mutually exclusive.) The probability of an outcome other than a complete pass is _____.

1/2

Let X be the number rolled on a fair six-sided die. Let Y = X2. The mean of Y is μY = ______. (Round to two decimal places.)

15.17

Suppose X, Y, and Z are random variables with means μX = 3, μY = 4, and μZ = 5. Find the mean of W = X + 2Y + Z.

Suppose pencil lengths, in inches, are randomly distributed with mean 6.5 inches. There are 2.54 centimeters in 1 inch. What is the mean pencil length in centimeters?

16.51

1/10 of the first names of students in a class start with the letter "R," 1/8 start with "S," and 1/5 start with "T." The probability that a randomly chosen student's first name will start with "R" or "T" is _____.

3/10

For the random variable X with probability density function f(x) = x/2 for 0 ≤ X ≤ 2 (and f(x) = 0 otherwise), P(X > 1) = _____.

3/4

Let F(x) denote the cumulative distribution function of the random variable X, the number of heads in two tosses of a fair coin. What is the value of F(1)?

3/4

Let F(x) denote the cumulative distribution function of the random variable X, the number of heads in two tosses of a fair coin. What is the value of F(1)? Multiple choice question.

3/4

4/5

If the jointly continuous random variables X and Y have the density function f(x,y) = (2/3)(2x + y) over the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (and 0 elsewhere), then P(0 ≤ x ≤ 1 and 0 ≤ y ≤ 0.5) = ______.

5/12

Select all the pairs of events that are mutually exclusive.

A: At least a 3 on a six-sided die B: A number less than 3 on a six-sided die A: An even number on a six-sided die B: A number divisible by 5 on a six-sided die

Select all pairs of events that are independent.

A: roll a 2 on a die B: flip tails on a coin A: draw a black ace from a full deck of playing cards B: draw a red card from a second full deck of playing cards

Match each property of the sample mean on the left with its value on the right.

Mean of X = μ Variance of X = σ2n Standard deviation of X = σ√n

The cumulative distribution function of a discrete random variable X is F(x) = _____.

P(X ≤ x)

A rope splicing shop's splices are weak with probability 1/1000 and ugly with probability 1/20; and 2/3 of weak splices are ugly. Find the probability that an ugly splice is weak.

P(ugly and weak) / P(ugly) P(ugly | weak) P(weak) / P(ugly) 1/75

Consider a random variable X with mean μX and standard deviation σX. Chebyshev's Inequality says that _____.

P(|X - μX| ≥ kσX) ≤ 1/k^2

Event A is rolling at least a 2 on a standard six-sided die. For which of the following definitions of event B would A and B be mutually exclusive?

Rolling a 1 on a die

Which of the following describe a permutation?

Select 3 students from 5 students to be president, vice president, and treasurer. Select 4 different digits to create a unique student ID number from the digits 0 to 9.

Which of the following describe a combination?

Select 3 winners to receive $50 each from 100 conference attendees. Elect 4 people from the members of a club to be on a committee.

If P(A) = 0.3, P(B) = 0.4, and P(B|A) = 0.4, then which of the following is true about events A and B?

The probability of event A remains the same whether or not the event B occurs. A and B are independent.

Probabilities of a continuous random variable are given by _____.

areas under its probability density function

If X is a random variable and a is a constant, then the mean of aX is μaX = _____.

aμX

If a first operation can be performed in m ways, and if for each of these ways a second operation can be performed in n ways, then the total number of ways to perform the sequence of the two operations is _____.

The number of ways to perform a sequence of operations is found by ________ the numbers of ways to perform each of the operations.

multiplying

The conditional probability mass function of Y given X for jointly discrete random variables X and Y is pY|X(y|x) = ______.

p(x,y)/pX(x)

A _____ is the selection of r objects from n objects when order matters.

permutation

The population variance of a discrete random variable X is given by _____.

∑x(x-μX)2 P(X=x)

If X is a discrete random variable with probability mass function p(x), and h(X) is a function of X, then the mean of h(X) is _____.

∑xh(x)p(x)

The mean of a discrete random variable X is μX = ____.

∑xxP(X = x)

If X and Y are jointly continuous random variables with density function f(x,y), and h(X,Y) is a function of X and Y, then the mean of h(X,Y) is _____.

∞−∞∫∞−∞h(x,y)f(x,y)dxdy

1/2

If X1 and X2 are independent random variables, each with variance σ2, find the variance of the variable Y = 12(X1 + X2).

1/2 σ^2

If the jointly continuous random variables X and Y have the density function f(x,y) = (2/3)(2x + y) over the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 (and 0 elsewhere), then the mean of h(X,Y) = XY is _____.

1/3

A board game player rolls two fair four-sided dice. Find the probability that the dice sum to 5.

1/4

For the random variable X with probability density function f(x) = 2x for 0 ≤ x ≤ 1 (and f(x) = 0 otherwise), P(X < 1/2) = _____.

1/4

Here are (mutually exclusive) outcomes of a quarterback throwing a pass with their associated probabilities: complete pass (1/2), incomplete pass (1/4), interception (1/8), or fumble (1/8). P(interception or fumble) = ______.

1/4

]For the random variable X with probability density function f(x) = 2x for 0 ≤ x ≤ 1 (and f(x) = 0 otherwise), P(X < 1/2) = _____.

1/4

In today's weather forecast, the probability of rain is 1/2, the probability of lightning is 3/10, and the probability of lightning given rain is 2/5. Find the probability of rain and lightning.

1/5

A personal identification Number (PIN) is created using four digits. How many different PINs are possible? Multiple choice question.

10000

In how many ways can a ten-person school board choose three of its members to serve on a committee for athletics? (Suppose the three committee positions are identical.)

120

A new community organization has 12 members. In how many ways can it elect a president, a vice president, and a secretary from among its members if no member may hold more than one office?

1320

The number of combinations of 5 objects chosen from a group of 8 objects is _____.

The number of permutations of 5 objects chosen from a group of 8 objects is _____.

6720

A deck of 52 playing cards consists of 4 suits (Clubs, Diamonds, Hearts, and Spades) of 13 cards each (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King). A randomly chosen card is drawn. The probability it is an even number (2, 4, 6, 8, or 10) or Club is _____. (Answer with a reduced fraction.) Multiple choice question.

7/13

For a random variable X with mean 5 and variance 4, the mean of X + 3 is

A deck of 52 playing cards consists of 4 suits (Club, Diamond, Heart, Spade) of 13 cards each (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King). A randomly-chosen card is drawn. Match each event to its probability.

Ace, 1/13 Club, 1/4 Ace or King, 2/13

The values of a random variable can be arranged in a (finite or infinite) sequence.

discrete

P(A), the that event A occurs, is the proportion of times that event A would occur in the long run if the experiment were repeated many times. (Answer with one word.)

probability

A variable assigns a numerical value to each outcome in a sample space.

random

If X is a random variable and a is a constant, the standard deviation of aX is σaX = _____.

|a|σX

Fahrenheit temperature F is related to Celsius temperature C by the formula F = (9/5)C + 32. Daily maximum Celsius temperatures in a city in June have mean μC = 27 and standard deviation σC = 3. For the corresponding Fahrenheit temperatures, match the property on the left with its value on the right. (Round to two decimal places.)

μF=80.6 σF=5.40 σ(?)F=29.16

For a random variable X and constants a and b, match each expression on the left with its equivalent on the right.

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