Deck 8: Continuous Probability Distributions

Question 1

Which of the following statements is correct regarding the percentile points of the F distribution?

A)F_0.10,10,20 = 1/F_0.90,10,20
B)F_0.90,10,20 = 1/F_0.10,20,10
C)F_0.90,10,20 = 1/F_0.90,20,10
D)F_0.10,10,20 = 1/F_0.10,20,10

Answer 1

F_0.90,10,20 = 1/F_0.10,20,10

Question 2

As the degrees of freedom approach infinity, the values of a Student t distribution approach those of a standard normal distribution.

Answer 2

True

Question 3

The value of $\chi$ ² with v degrees of freedom such that the area to its right under the chi-squared curve is equal to A is denoted by $\chi _ { A , v } ^ { 2 }$ , while $\chi _ { 1 - A , v } ^ { 2 }$
denotes the value such that the area to its left is A.

Answer 3

True

Question 4

The Student t distribution:

A)is symmetrical.
B)approaches the normal distribution as the degrees of freedom increase.
C)has more area in the tails than the standard normal distribution does.
D)All of these choices are true.

Question 5

The variance of a Student t distribution approaches zero as the degrees of freedom approaches infinity.

Question 6

Which of the following statements is false?

A)The chi-squared distribution is positively skewed.
B)All the values of the chi-squared distribution are non-negative.
C)The shape of the chi-squared distribution depends on its degrees of freedom.
D)All of these choices are true.

Question 7

The variance of a Student t random variable with v degrees of freedom (v > 2) is always greater than 1.

Question 8

The expected value of the Student t distribution is zero.

Question 9

Like that of the Student t distribution, the shape of the chi-squared distribution depends on its number of degrees of freedom.

Question 10

We define $F _ { A } , v _ { 1 } , v _ { 2 }$ as the value of the F with v₁ and v₂ degrees of freedom such that the area to its right under the F curve is A, while $F _ { 1 } - A _ { 1 } v _ { 1 } , v _ { 2 }$ is defined as the value such that the area to its left is A.

Question 11

The value of a chi-squared distribution with 8 degrees of freedom such that the area to its left is 0.95 is 2.73.

Question 12

The Student t distribution looks similar in shape to a standard normal distribution, except it is not as widely spread.

Question 13

What number corresponds to t_0.05,10?

A)1.812
B)1.372
C)2.228
D)1.833

Question 14

The value of a chi-squared distribution with 5 degrees of freedom such that the area to its left is 0.10 is 1.61.

Question 15

The Student t distribution with parameter v = 2 has a mean E(t) equal to:

A)0
B)1
C)2
D)None of these choices.

Question 16

The value of A such that P(-A $\le$ t $\le$ A) = 0.95, where the degrees of freedom are 20, is 2.086.

Question 17

The Student t distribution with parameter v = 4 has a variance V(t) equal to:

A)4
B)0
C)2
D)1

Question 18

The value of an F distribution with v₁ = 5 and v₂ = 10 degrees of freedom such that the area to its left is 0.95 is 3.33.

Question 19

The variance of a $\chi$ ² distribution is twice the value of its mean.

Question 20

The value of an F distribution with v₁ = 6 and v₂ = 9 degrees of freedom such that the area to its right is 0.05 is 3.37.

Question 21

If P(t > t_.01,v) = 2.50, then the number of degrees of freedom v is:

A)20
B)21
C)22
D)23

Question 22

The shape of a X² distribution is ____________________.

Question 23

What number corresponds to F_.025,3,5?

A)14.88
B)7.76
C)12.06
D)5.41

Question 24

Suppose X has an F distribution. Which of the following is true?

A)f(x) is symmetrical.
B)All the values of X are non-negative.
C)The mean of X is zero.
D)All of these choices.

Question 25

Suppose X has a chi-squared distribution with 10 degrees of freedom. The mean of X is:

A)10
B)9
C)20
D)None of these choices.

Question 26

The mean and variance of a X² distribution approach ____________________ as the degrees of freedom increase.

Question 27

What number corresponds to $x _ { 0.05,12 } ^ { 2 }$ ?

A)28.30
B)26.22
C)21.00
D)5.23

Question 28

The F distribution has two parameters called degrees of freedom, $\nu$ ₁ and $\nu$ ₂. We call $\nu$ ₁ the ____________________ degrees of freedom, and we call $\nu$ ₂ the ____________________ degrees of freedom.

Question 29

Which of the following distributions can take on negative values?

A)Student t
B) $\chi$ ²
C)F
D)None of these choices.

Question 30

Which of the following has a mean and variance that depend on degrees of freedom?

A)Student t
B) $\chi$ ²
C)F
D)All of these choices are true.

Question 31

The variance of a Student t distribution approaches ____________________ as the degrees of freedom increase to infinity.

Question 32

The mean of a Student t distribution is ____________________.

Question 33

A X² distribution with 5 degrees of freedom has a mean of ____________________ and a variance of ____________________.

Question 34

If $P \left( X ^ { 2 } > X _ { 025 , v } ^ { 2 } \right) = 14.4$ , then the number of degrees of freedom v is:

A)5
B)6
C)7
D)8

Question 35

The shape of the ____________________ distribution is similar to a normal distribution, except it has more area in the tails.

Question 36

The number of parameters for an F distribution is:

A)1
B)2
C)0
D)None of these choices.

Question 37

Which of the following distributions is not skewed?

A)Student t
B) $\chi$ ²
C)F
D)All of these distributions are skewed.

Question 38

Suppose X has a chi-squared distribution with 10 degrees of freedom. The variance of X is:

A)20
B)10
C)9
D)None of these choices.

Question 39

What number corresponds to F_0.95,4,8?

A)6.040
B)3.840
C)0.260
D)0.166

Question 40

For values of degrees of freedom greater than 100, the X² distribution can be approximated by a(n) ____________________ distribution.

Question 41

Use the t-table to find the following probabilities.
a.P(t₈ > 2.306)
b.P(t₈₀ > 2.639)
c.P(t₂₄ > 1.711)
d.P(t₃₅ > 1.306)

Question 42

Suppose you have an F distribution with degrees of freedom 10, 20.
a.Find the mean
b.Find the variance.
c.Find the standard deviation.

Question 43

The mean and standard deviation of an exponential random variable are equal to each other.

Question 44

If the random variable X is exponentially distributed with parameter $\lambda$ = 0.05, then the variance of X, $\sigma$ ² = V(X) = 20.

Question 45

Use the F table to find the following values of F.
a.F_.99,12,20
b.F_.95,20,40
c.F_.975,5,15
d.F_.99,8,30

Question 46

The shape of an F distribution is ____________________.

Question 47

As the ____________________ degrees of freedom increase, the mean of the F distribution approaches 1.

Question 48

Suppose you have a Student t distribution with 20 degrees of freedom.
a.Find the mean.
b.Find the variance.
c.Find the standard deviation.

Question 49

Use the F table to find the following probabilities.
a.P(F_6,14 > 2.85)
b.P(F_20,60 > 2.20)
c.P(F_12,25 > 2.51)
d.P(F_15,30 > 2.01)

Question 50

The exponential distribution is suitable to model the length of time that elapses before the first telephone call is received by a switchboard.

Question 51

If the random variable X is exponentially distributed and the parameter of the distribution $\lambda$ = 4, then P(X $\le$ 1) = 0.25.

Question 52

If the random variable X is exponentially distributed with parameter $\lambda$ = 5, then the variance of X, $\sigma$ ² = V(X) = 0.04.

Question 53

Suppose you have a X² distribution with 20 degrees of freedom.
a.Find the mean.
b.Find the variance.
c.Find the standard deviation.

Question 54

If the mean of an exponential distribution is 2, then the value of the parameter $\lambda$ is 2.0.

Question 55

What happens to the shape, mean, and variance of a Student t distribution as the degrees of freedom increase?

Question 56

The mean and the variance of an exponential distribution are equal to each other.

Question 57

What happens to the shape, mean, and variance of a X² distribution as the degrees of freedom increase?

Question 58

Use the F table to find the following values of F.
a.F_.01,12,20
b.F_.05,20,40
c.F_.025,5,15
d.F_.01,8,30

Question 59

Use the t-table to find the following values of t.
a.t_.10,9
b.t_.10,20
c.t_.025,82
d.t_.05,196

Question 60

In the exponential distribution, X takes on an infinite number of possible values in the given range.

Question 61

The mean of an exponential random variable is ____________________ the median.

Question 62

The exponential density function f(x):

A)is bell-shaped.
B)is symmetrical.
C)approaches infinity as x approaches zero.
D)approaches zero as x approaches infinity.

Question 63

If X has an exponential distribution, the possible values of X are from ____________________ to infinity.

Question 64

A random variable with density function 0.01e^-x/100 for x $\ge$ 0 has an exponential distribution whose mean is ____________________.

Question 65

If the random variable X is exponentially distributed, then the mean of X will be:

A)greater than the median.
B)less than the median.
C)equal to the median.
D)Cannot tell; the answer depends on what $\lambda$ is.

Question 66

An exponential random variable is an example of a(n) ____________________ random variable.

Question 67

If the random variable X is exponentially distributed with parameter $\lambda$ = 4, then the probability P(X $\le$ 0.25), up to 4 decimal places, is

A)0.6321
B)0.3679
C)0.2500
D)None of these choices.

Question 68

Which of the following can have an exponential distribution?

A)Time between phone calls coming in to a technical support desk.
B)Time until the first customer arrives at the bank in the morning.
C)Lifetime of a new battery.
D)All of these choices are true.

Question 69

If the random variable X is exponentially distributed with parameter $\lambda$ = 3, then the probability P(X $\ge$ 2) equals:

A)0.3333
B)0.5000
C)0.6667
D)0.0025

Question 70

If the parameter of an exponential distribution is 1, then which of the following is not true?

A)The density function is e^-x for x $\ge$ 0.
B)The mean is equal to 1.
C)The standard deviation and variance are both equal to 1.
D)All of these choices are true.

Question 71

If X has an exponential distribution with parameter $\lambda$ , then f(0) = ____________________.

Question 72

If the mean of an exponential distribution is 2, then the value of the parameter $\lambda$ is

A)0
B)2.0
C)0.5
D)1.0

Question 73

If the random variable X is exponentially distributed with parameter $\lambda$ = 0.05, then the probability P(X < 5) = .2865.

Question 74

The shape of the density function for an exponential distribution is ____________________.

Question 75

A random variable with density function e^-x for x $\ge$ 0 has an exponential distribution with $\lambda$ = ____________________.

Question 76

Which of the following is not true for an exponential distribution with parameter $\lambda$ ?

A) $\mu$ = 1/ $\lambda$
B) $\sigma$ = 1/ $\lambda$
C)The Y-intercept of f(x) is $\lambda$ .
D)All of these choices are true.

Question 77

If the random variable X is exponentially distributed with parameter $\lambda$ = 0.05, then the probability P(X > 20) = 0.3679.

Question 78

A random variable with density function e^-x for x $\ge$ 0 has an exponential distribution whose mean is ____________________.

Question 79

If the random variable X is exponentially distributed with parameter $\lambda$ = 2, then the probability that X is between 1 and 2 equals the probability that X is between 2 and 3.

Question 80

If the random variable X is exponentially distributed with parameter $\lambda$ = 1.5, then the probability P(2 $\le$ X $\le$ 4), up to 4 decimal places, is

A)0.6667
B)0.0473
C)0.5000
D)0.2500