# Exam P Practice Test

95 Questions | Total Attempts: 288  Settings  Taken from the four 1/P Exams from The Infinite Actuary.

• 1.
75% of the customers of ACME Mutual Insurance have auto insurance, and 40% have homeowners insurance. What is the maximum possible probability that a randomly selected customer with auto insurance does not have homewoners insurance?
• A.

20%

• B.

40%

• C.

60%

• D.

80%

• E.

100%

• 2.
Suppose  is a normal random variable with  and coefficient of variation 3. Find .
• A.

0.05

• B.

0.34

• C.

0.57

• D.

0.66

• E.

0.95

• 3.
The probability that Rafael Nadal wins a tennis match in straight sets is 70%. Assuming that the outcome of each match is independent, what is the probability that in his next 7 matches he will win in straight sets at least 5 times?
• A.

0.13

• B.

0.33

• C.

0.44

• D.

0.65

• E.

0.96

• 4.
and  have joint density given by  Find .
• A.

0.7

• B.

0.9

• C.

1.2

• D.

1.4

• E.

1.6

• 5.
The joint cdf of  is given by . for . Find .
• A.

0.05

• B.

0.17

• C.

0.19

• D.

0.41

• E.

0.61

• 6.
Let  be the number of rolls of a fair die before getting a 6 and let  be the number of rolls before the first even number. Find .
• A.

5

• B.

6

• C.

7

• D.

8

• E.

9

• 7.
Suppose  and  are bivariate random variables with  and . If  find .
• A.

0.40

• B.

0.45

• C.

0.53

• D.

0.55

• E.

0.60

• 8.
Suppose that I roll two independent dice one red and one blue. Let  be the event that the blue die is even  the event that the red die is even and  the event that the sum is even. Which of the following is true?
• A.

None of them are independent.

• B.

A and B are pairwise independent, but neither is pairwise independent of C.

• C.

A and C are pairwise independent, as are B and C, but A and B are not pairwise independent.

• D.

All three pairs are pairwise independent, but it is not true that all three are mutually independent.

• E.

All three are mutually independent.

• 9.
For  the joint density of  is given by . Find .
• A.

0.75

• B.

0.78

• C.

0.81

• D.

0.84

• E.

0.87

• 10.
A student who is taking a 30-question multiple choice test knows the answer to 24 of the questions. Whenever the student doesn't know the answer to a question, he chooses uniformly from one of the 5 choices. Given that the student gets a randomly chosen question right, what is the probability that the student guessed on the question?
• A.
• B.
• C.
• D.
• E.
• 11.
If   and  are i.i.d. exponential random variables with mean 3 what is ?
• A.

27

• B.

54

• C.

81

• D.

108

• E.

135

• 12.
Suppose that  and  are independent Poisson random variables with  and . Find .
• A.

0.17

• B.

0.21

• C.

0.25

• D.

0.29

• E.

0.34

• 13.
Let  be a randomly chosen integer with . What is the probability that  is not divisible by 7 11 or 13?
• A.

0.66

• B.

0.69

• C.

0.72

• D.

0.75

• E.

0.78

• 14.
Insurance losses  in a given year have a lognormal distribution with  where  is a normal random variable with mean 3.9 and standard deviation 0.8. If a \$100 deductible and a \$50 benefit limit are imposed what is the probability that the insurance company will pay the benefit limit given that a loss exceeds the deductible?
• A.

0.10

• B.

0.27

• C.

0.43

• D.

0.66

• E.

0.88

• 15.
A fair 6-sided die is rolled 1,000 times. Using a normal approximation with a continuity correction, what is the probability that the number of 3's that are rolled is greater than 150 and less than 180?
• A.

0.78

• B.

0.81

• C.

0.84

• D.

0.88

• E.

0.95

• 16.
Four red dice and six blue dice are rolled. Assuming that all ten dice are fair six-sided dice, and rolls are independent, what is the probability that exactly three of the red dice are even, and exactly two of the blue dice come up ones?
• A.

0.05

• B.

0.10

• C.

0.16

• D.

0.21

• E.

0.27

• 17.
A life insurance company classifies its customers as being either high or low risk. If 20% of the customers are high risk, and high risk customers are three times as likely as low risk customers to file a claim, what percentage of claims that are filed come from high risk customers?
• A.

30%

• B.

37%

• C.

43%

• D.

54%

• E.

60%

• 18.
Suppose that  are random variables with  and . If  for  what is  where .
• A.

0

• B.

100

• C.

1,000

• D.

5,050

• E.

10,000

• 19.
The moment generating function of  is . Find .
• A.

1

• B.

2

• C.

3

• D.

4

• E.

5

• 20.
The cdf of a random variable  satisfies for . Find .
• A.

0.22

• B.

0.36

• C.

0.51

• D.

0.64

• E.

0.78

• 21.
The density of  is proportional to  for  and is 0 otherwise. Find the 80th percentile of .
• A.

0.9

• B.

1.3

• C.

1.8

• D.

2.3

• E.

2.8

• 22.
If  is a Poisson random variable with  then what is the probability that  will be within 1 standard deviation of ?
• A.

0.08

• B.

0.29

• C.

0.47

• D.

0.63

• E.

0.81

• 23.
The joint density of  and  is for  and . Find .
• A.

1.1

• B.

1.3

• C.

1.5

• D.

1.7

• E.

1.9

• 24.
Suppose that  are i.i.d. uniform random variables on the interval . Let  denote the average of  through  and let  and  denote the standard deviation and mean of . Find the probability that the minimum and maximum of  both differ from  by less than .
• A.

0.00001

• B.

0.00032

• C.

0.00057

• D.

0.00083

• E.

0.00115

• 25.
Find the mode of a Poisson random variable with mean 1.5.
• A.

0

• B.

0.5

• C.

1

• D.

1.5

• E.

2

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