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Question : The life (in months) of an automobile battery has a probability density function : 2151846

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Decide whether or not the function is a probability density function on the indicated interval.

1) f(x) = (3/98)x^{2}; [3, 5]

A) Yes

B) No

2) f(x) = (4/81)x^{3}; [0, 3]

A) Yes

B) No

3) f(x) = (1/3)x - (1/6); [3, 4]

A) Yes

B) No

4) f(x) = (1/9)x - (1/18); [2, 5]

A) No

B) Yes

5) f(x) = (3/63)x^{2}; [1, 4]

A) No

B) Yes

6) f(x) = (4/74)x^{3}; [0, 3]

A) Yes

B) No

7) f(x) = 3x^{2}; [-2, 2]

A) Yes

B) No

8) f(x) = 2x; [-2, √(5)]

A) No

B) Yes

9) f(x) = (3x^{2}/16); [-2, 2]

A) No

B) Yes

10) f(x) = (2/109)x^{2} - (2/109)x + (1/654); [0, 6]

A) No

B) Yes

Find a value of k that will make f a probability density function on the indicated interval.

11) f(x) = kx; [0, 5]

A) (2/25)

B) (2/5)

C) (1/25)

D) (1/5)

12) f(x) = kx^{2}; [0, 3]

A) (3/26)

B) (1/27)

C) (2/9)

D) (1/9)

13) f(x) = kx^{2}; [-1, 4]

A) (3/65)

B) (1/16)

C) (3/64)

D) (1/21)

14) f(x) = kx^{1/2}; [1, 9]

A) (1/56)

B) (3/54)

C) (3/17)

D) (3/52)

15) f(x) = kx^{2}; [1, 2]

A) (3/7)

B) (1/3)

C) (3/8)

D) (1/2)

16) f(x) = kx; [2, 4]

A) (1/6)

B) (1/8)

C) (1/12)

D) (1/16)

17) f(x) = kx; [0, 6]

A) (1/35)

B) (1/36)

C) (1/9)

D) (1/18)

18) f(x) = kx^{2}; [1, 3]

A) (3/28)

B) (3/26)

C) (1/13)

D) (1/9)

19) f(x) = kx^{3}; [0, 2]

A) (1/4)

B) (1/15)

C) (4/15)

D) (1/8)

20) f(x) = kx^{3}; [1, 2]

A) (3/16)

B) (4/17)

C) (1/4)

D) (4/15)

Find the cumulative distribution function for the given probability density function.

21) f(x) = (1/5)x - (1/10); [2, 4]

A) F(x) = (x^{2} -x - 4/10), 2 ≤x ≤ 4

B) F(x) = (x^{2} -x + 2/10), 2 ≤x ≤ 4

C) F(x) = (x^{2} -x - 2/10), 2 ≤x ≤ 4

D) F(x) = (x^{2} -x - 2/5), 2 ≤x ≤ 4

22) f(x) = (3/56)x^{2}; [2, 4]

A) F(x) = (x^{3} - 2/56), 2 ≤x ≤ 4

B) F(x) = (x^{3} + 8/56), 2 ≤x ≤ 4

C) F(x) = (x^{3}/56), 2 ≤x ≤ 4

D) F(x) = (x^{3} - 8/56), 2 ≤x ≤ 4

23) f(x) = (3/52)x^{1/2}; [1, 9]

A) F(x) = (x^{3/2}/26), 1 ≤x ≤ 9

B) F(x) = (x^{1/2} - 1/26), 1 ≤x ≤ 9

C) F(x) = (x^{3/2} - 1/26), 1 ≤x ≤ 9

D) F(x) = (x^{3/2} + 1/26), 1 ≤x ≤ 9

Find the indicated probability.

24) f(x) = (1/2)(1 +x)^{-3/2}; [0, ∞), P(x ≥ 24)

A) (1/5)

B) (1/√(24))

C) - (1/5)

D) - (1/√(24))

25) f(x) = e^{-x}; [0, ∞), P(x ≥ 7)

A) 00.9991

B) 0

C) 0.0009

D) The function f(x) is not a probability density function.

26) f(x) = (1/5)e^{-x/5}; [0, ∞), P(1 ≤x ≤ 6)

A) 0.0518

B) 0.5175

C) 0.0690

D) 0.1035

27) f(x) = (6/(x + 6)^{2}); [0, ∞), P(2 ≤x ≤ 5)

A) 0.1364

B) 0.2045

C) 0.4773

D) The function f(x) is not a probability density function.

28) f(x) = {(x^{3}/12) if 0 ≤x ≤2

(16/3x^{3}) ifx > 2, P(1 ≤x ≤ 4)

A) (35/48)

B) (37/48)

C) (5/6)

D) (13/16)

Solve the problem.

29) The life (in months) of an automobile battery has a probability density function defined by f(x) = (1/4)e^{-x/4} for x in [0, ∞). Find the probability that the life of a randomly selected battery is greater than 5 years.

A) 0.0716

B) 0.1784

C) 0.7135

D) 0.2865

30) The time between major earthquakes in the Alaska panhandle region is a random variable with probability density function f(x) = (1/700)e^{-x/700} for x in [0, ∞), where t is measured in days. Find the probability that the time between a major earthquake and the next one is less than 200 days.

A) 0.0011

B) 0.0004

C) 0.7515

D) 0.2485

31) The time of a telephone call (in minutes) to a certain town is a continuous random variable with a probability density function defined by f(x) = 2x^{-3} forx ≥ 1. Find the probability that the call lasts between 1 and 3 minutes.

A) 0.8889

B) 0.1111

C) 1.1111

D) 0.7639

32) The time of a telephone call (in minutes) to a certain town is a continuous random variable with a probability density function defined by f(x) = 7x^{-8} forx ≥ 1. Find the probability that the call lasts more than 3 minutes.

A) 0.0009

B) 0.8745

C) 0.0005

D) 0.9995

33) The time to failure t, in hours, of a certain machine can often be assumed to be exponentially distributed with probability density function

f(t) = (1/82)e^{-t/82}, 0 ≤ t < ∞,

What is the probability that a failure will occur in 45 hours or less?

A) 0.4223

B) 0.5068

C) 0.3379

D) 0.3168