Exercises

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Jun 01'22

The random variable [math]X[/math] has a distribution with the following density function:

[[math]] f_X(x) = \begin{cases} \theta^{-1} e^{-x/\theta} \,\, x\geq 0 \\ 0 \,\, \textrm{otherwise} \end{cases} [[/math]]

where [math]\theta [/math] is unknown. If the median of the distribution equals 250, determine the standard deviation of [math]X[/math].

19
250
300
361
366

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May 01'23

A random variable [math]X[/math] has the cumulative distribution function

[[math]] F(x) = \begin{cases} 0, \, x \lt 1 \\ \frac{x^2-2x +2}{2}, \, 1 \leq x \lt 2 \\ 1, \, x \geq 2. \end{cases} [[/math]]

Calculate the variance of [math]X[/math].

7/72
1/8
5/36
4/3
23/12

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May 01'23

The warranty on a machine specifies that it will be replaced at failure or age 4, whichever occurs first. The machine’s age at failure, [math]X[/math], has density function

[[math]] f(x) = \begin{cases} \frac{1}{5}, \, 0\lt x \lt 5 \\ 0, \, \textrm{Otherwise.} \end{cases} [[/math]]

Let [math]Y[/math] be the age of the machine at the time of replacement. Calculate the variance of [math]Y[/math].

1.3
1.4
1.7
2.1
7.5

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May 07'23

A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 with a variance of 260. A tax of 20% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 20% more expensive). Calculate the variance of the annual cost of maintaining and repairing a car after the tax is introduced.

208
260
270
312
374

ABy Admin

May 07'23

An airport purchases an insurance policy to offset costs associated with excessive amounts of snowfall. For every full ten inches of snow in excess of 40 inches during the winter season, the insurer pays the airport 300 up to a policy maximum of 700. The following table shows the probability function for the random variable [math]X[/math] of annual (winter season) snowfall, in inches, at the airport.

Inches	[0,20)	[20,30)	[30,40)	[40,50)	[50,60)	[60,70)	[70,80)	[80,90)	[90,inf)
Probability	0.06	0.18	0.26	0.22	0.14	0.06	0.04	0.04	0.00

Calculate the standard deviation of the amount paid under the policy.

134
235
271
313
352

ABy Admin

May 08'23

An insurance company has an equal number of claims in each of three territories. In each territory, only three claim amounts are possible: 100, 500, and 1000. Based on the company’s data, the probabilities of each claim amount are:

	Total Amount
	100	500	1000
Territory 1	0.90	0.08	0.02
Territory 2	0.80	0.11	0.09
Territory 3	0.70	0.20	0.10

Calculate the standard deviation of a randomly selected claim amount.

254
291
332
368
396

ABy Admin

May 09'23

The number of claims [math]X[/math] on a health insurance policy is a random variable with [math]\operatorname{E}[ X^2 ] = 61[/math] and [math]\operatorname{E}[( X -1)^2 ] = 47 [/math] .

Calculate the standard deviation of the number of claims.

2.18
2.40
7.31
7.50
7.81

ABy Admin

May 07'23

A probability distribution of the claim sizes for an auto insurance policy is given in the table below:

Claim Size	Probability
20	0.15
30	0.10
40	0.05
50	0.20
60	0.10
70	0.10
80	0.30

Calculate the percentage of claims that are within one standard deviation of the mean claim size.

45%
55%
68%
85%
100%

ABy Admin

Jun 24'24

A die is loaded so that the probability of a face coming up is proportional to the number on that face. The die is rolled with outcome [math]X[/math]. Find [math]Var(X)[/math].

1.85
2.22
2.72
3.15
3.62

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.

ABy Admin

Jun 24'24

A number is chosen at random from the integers 1, 2, 3,...,10. Let [math]X[/math] be the number chosen. Determine [math]\operatorname{Var}(X)[/math].

8.25
8.5
8.75
9
9.25

References

Doyle, Peter G. (2006). "Grinstead and Snell's Introduction to Probability" (PDF). Retrieved June 6, 2024.