ENVE 3510: Concepts 1
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?
40
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 = ______.
5
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?
0
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.
24
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 _____.
4
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.
16
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
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 a letter other than "T" is _____. (Answer with a reduced fraction.)
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 _____.
mn
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
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). Let A be the event consisting of an incomplete pass, interception or a fumble. Then P(A) is _____. (Answer with a reduced fraction.)
1/2
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 _____. Multiple choice question.
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 _____.
56
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
8
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.
μaX+b=aμX + b σ2aX+b=a2σX2 σaX+b=|a|σX