**Q1 An urn contains four balls numbered 1, 2,3 and 4. If two balls are drawn from the urn at random and Z is the sum of the numbers on the two balls drawn, find the probability distribution of Z.**

f(x=4,6 and 7)=1/6 and f(x=3,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(x=2,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(5)=1/3

f(x=3,4)=1/6 and f(x=5,6,7)=1/3

**Q2 Let X be a continuous random variable with cdf $$F(x)=x/4[1+ln(4/x)] \; for \; 0 < x \leq 4 $$. What is $$P(1 \leq X \leq 3)$$**

0.244

0.369

0.451

0.512

**Q3 Let X be a continuous random variable with cdf $$F(x)=x/4[1+ln(4/x)] \; for \; 0 < x \leq 4 $$. Find the pdf of X**

$$ f(x)=0.212 -.25x \; for \; 0 <x <4 $$

$$ f(x)=2.612 -.31e^{x} \; for \; 0 <x <4 $$

$$ f(x)=0.612 -.21e^{2x} \; for \; 0 <x <4 $$

$$ f(x)=0.3466 -.25ln(x) \; for \; 0 <x <4 $$

**Q4 Determine k so that $$f(x,y)=kx(x-y) \; for \; 0 <x <1, \; -x < y < x , and f(x,y)= 0 \; elsewhere $$ can serve as a joint probability density.**

2

1

3

5

**Q5 The probability density of the random variable Z is given by $$f(z)=kze^{-z^{2}} \; for z > 0 \; and \; f(z)=0 \; elsewhere $$ . what is the distribution function of Z?**

$$1- 2e^{z^{-2}} \; for \; z > 0$$

$$e^{-z^{3}} \; for \; z > 0$$

$$1- e^{-z^{2}} \; for \; z > 0$$

$$1+ 3e^{-z^{-2}} \; for \; z > 0$$

**Q6 If the joint probability distribution of three discrete random variables X, Y, Z is given by $$f(x,y,z)=\frac {(x +y)z}{63} \; for \; x=1,2; \; y=1,2,3; \; z=1,2 $$. Find $$P(X=2, Y+Z \leq{3})$$**

12/47

15/67

5/18

13/63

**Q7 Given the joint probability density $$f(x,y)= \frac {2(x+2y)}{3} \; for \; 0 <x <1, \; 0 <y <1 $$ and $$f(x,y)= 0; \; elsewhere $$. Find the marginal density of Y**

$$ h(y)= \frac{1+y}{2} \; for \; 0 <y <1 $$ and $$h(y)= 0; \; elsewhere $$

$$ h(y)= \frac{1+4y}{3} \; for \; 0 <y <1 $$ and $$h(y)= 0; \; elsewhere $$

$$ h(y)= \frac{3y+4}{2} \; for \; 0 <y <1 $$ and $$h(y)= 0; \; elsewhere $$

$$ h(y)= \frac{7y}{5} \; for \; 0 <y <1 $$ and $$h(y)= 0; \; elsewhere $$

**Q8 The number of minutes that a flight from Abuja to Kaduna is early or late is a random variable whose probability density is given by $$f(x)=\frac {36-x^{2}} {288}$$, for -6 <x <6 and f(x)=0, elsewhere. Where negative values are indicative of flight****�****??s being early and positive values are indicative of its being late. Find the probability that one of these flights will be anywhere from 1 to 3 minutes early**

12/331

95/432

51/552

13/121

**Q9 For what values of c can $$f(x)=\frac {c} {x}$$ serve as the values of the probability distribution of a random with countably infinite range x=0,1,2,****�****?****��****?****��****?****��****?****��****?****�****..?**

no value

2

6

1

**Q10 The probability distribution of V, the weekly number of mangoes that will freely from the tree at a certain region is given by g(0) = 0.40, g(1) = 0.30, g(2) = 0.20 and g(3) = 0.10 , find the probability that there will be at least 2 mangoes that will freely in any one week**

0.4

0.5

0.1

0.3

**Q11 An urn contains four balls numbered 1, 2,3 and 4. If two balls are drawn from the urn at random and Z is the sum of the numbers on the two balls drawn, find the probability distribution of Z.**

f(x=4,6 and 7)=1/6 and f(x=3,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(x=2,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(5)=1/3

f(x=3,4)=1/6 and f(x=5,6,7)=1/3

**Q12 If X has the probability density $$f(x)= e^{-x} \; for \; x >0, \; f(x)= 0 \; elsewhere.$$ find the expected value of $$ g(X)=e^{3X/4}$$**

3

5

4

6

**Q13 If the joint probability density of X and Y is given f(x,y)=2 \; for \; x >0, \; y>0, x+y <1 \; and f(x,y)= 0 \; elsewhere $$ . find P(X <=1/2 , Y <= ½)**

0.42

0.62

0.7

0.5

**Q14 Determine k so that $$f(x,y)=kx(x-y) \; for \; 0 <x <1, \; -x < y < x , and f(x,y)= 0 \; elsewhere $$ can serve as a joint probability density.**

2

1

3

5

**Q15 The probability density of the random variable Z is given by $$f(z)=kze^{-z^{2}} \; for z > 0 \; and \; f(z)=0 \; elsewhere $$ . what is the distribution function of Z?**

$$1- 2e^{z^{-2}} \; for \; z > 0$$

$$e^{-z^{3}} \; for \; z > 0$$

$$1- e^{-z^{2}} \; for \; z > 0$$

$$1+ 3e^{-z^{-2}} \; for \; z > 0$$

**Q16 Given a random variable X, and constants a, b. which of the following is/are true . (i) $$E(aX + b)=aE(X)$$ (ii) $$E(aX + b)=aE(X)+b$$ (iii) $$Var(aX + b)= a^{2}Var(X)$$, (iv) $$Var(aX + b)= a^{2}Var(X) + b^{2}$$,**

(ii) and (iii)

(i) and (iv)

(ii) and (iv)

only (ii)

**Q17 If X has the probability density $$f(x)=e^{-x} \; for \; x > 0 \; and \; f(x)=0 \; elsewhere$$. Find the expected value of $$g(X)=e^{3X/4} $$.**

6

8

2

4

**Q18 If joint probability density of X and Y is given by $$f(x,y)=\frac {2(x+y)}{7} \; for \; 0 <x <1, \; 1 <y <2 \; and \; f(x,y)=0, \; elsewhere $$. Find the expected value of $$g(X,Y)= \frac {X}{Y^{3}} $$**

14/65

15/84

17/93

23/72

**Q19 The useful life (in hours) of a certain kind of vacuum tubes is a random variable having the probability density $$f(x)=\frac{20,000}{(x + 100)^{3}} \; for \; x >0, $$, and $$f(x)= 0; \; elsewhere $$. If three of these tubes operative independently, find the joint probability density of $$ X_{1}, \; X_{2}, \; X_{3} $$, representing the lengths of their useful lives.**

$$ \frac {(10,000)} {(x_{1}+100)(x_{2}+100)(x_{3}+100)}$$

$$ \frac {(10,000)^{2}} {(x_{1}+100)^{2}(x_{2}+100)^{2}(x_{3}+100)^{2}}$$

$$ \frac {(20,000)^{3}} {(x_{1}+100)^{3}(x_{2}+100)^{3}(x_{3}+100)^{3}}$$

$$ \frac {(3,000)} {(x_{1}+100)(x_{2}+100)(x_{3}+100)}$$

**Q20 If X is the amount of money that a salesperson spends on gasoline during a day and Y is the corresponding amount of money for which he or she is reimbursed, the joint density of two random variables is given by $$f(x,y)=\frac{1}{25} \left ( \frac {20-x}{x} \right ); \; for \; 10 < x <20, \; x/2 <y <x $$, and $$f(x,y)= 0; \; elsewhere $$. Find the conditional density of Y given X = 12.**

$$h(y|12)= 1/6 \; for \; 6 < x < 12, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 2/3 \; for \; 1 < x < 2, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 1/3 \; for \; 1 < x < 6, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 2/9 \; for \; 0 < x < 12, \; h(y|12)= 0; \; elsewhere $$

**Q21 An urn contains four balls numbered 1, 2,3 and 4. If two balls are drawn from the urn at random and Z is the sum of the numbers on the two balls drawn, find the probability distribution of Z.**

f(x=4,6 and 7)=1/6 and f(x=3,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(x=2,5)=1/3

f(x=3,4,6 and 7)=1/6 and f(5)=1/3

f(x=3,4)=1/6 and f(x=5,6,7)=1/3

**Q22 If X has the probability density $$f(x)= e^{-x} \; for \; x >0, \; f(x)= 0 \; elsewhere.$$ find the expected value of $$ g(X)=e^{3X/4}$$**

3

5

4

6

**Q23 If the joint probability density of X and Y is given f(x,y)=2 \; for \; x >0, \; y>0, x+y <1 \; and f(x,y)= 0 \; elsewhere $$ . find P(X <=1/2 , Y <= ½)**

0.42

0.62

0.7

0.5

**Q24 Determine k so that $$f(x,y)=kx(x-y) \; for \; 0 <x <1, \; -x < y < x , and f(x,y)= 0 \; elsewhere $$ can serve as a joint probability density.**

2

1

3

5

**Q25 The probability density of the random variable Z is given by $$f(z)=kze^{-z^{2}} \; for z > 0 \; and \; f(z)=0 \; elsewhere $$ . what is the distribution function of Z?**

$$1- 2e^{z^{-2}} \; for \; z > 0$$

$$e^{-z^{3}} \; for \; z > 0$$

$$1- e^{-z^{2}} \; for \; z > 0$$

$$1+ 3e^{-z^{-2}} \; for \; z > 0$$

**Q26 Given a random variable X, and constants a, b. which of the following is/are true . (i) $$E(aX + b)=aE(X)$$ (ii) $$E(aX + b)=aE(X)+b$$ (iii) $$Var(aX + b)= a^{2}Var(X)$$, (iv) $$Var(aX + b)= a^{2}Var(X) + b^{2}$$,**

(ii) and (iii)

(i) and (iv)

(ii) and (iv)

only (ii)

**Q27 If X has the probability density $$f(x)=e^{-x} \; for \; x > 0 \; and \; f(x)=0 \; elsewhere$$. Find the expected value of $$g(X)=e^{3X/4} $$.**

6

8

2

4

**Q28 If joint probability density of X and Y is given by $$f(x,y)=\frac {2(x+y)}{7} \; for \; 0 <x <1, \; 1 <y <2 \; and \; f(x,y)=0, \; elsewhere $$. Find the expected value of $$g(X,Y)= \frac {X}{Y^{3}} $$**

14/65

15/84

17/93

23/72

**Q29 The useful life (in hours) of a certain kind of vacuum tubes is a random variable having the probability density $$f(x)=\frac{20,000}{(x + 100)^{3}} \; for \; x >0, $$, and $$f(x)= 0; \; elsewhere $$. If three of these tubes operative independently, find the joint probability density of $$ X_{1}, \; X_{2}, \; X_{3} $$, representing the lengths of their useful lives.**

$$ \frac {(10,000)} {(x_{1}+100)(x_{2}+100)(x_{3}+100)}$$

$$ \frac {(10,000)^{2}} {(x_{1}+100)^{2}(x_{2}+100)^{2}(x_{3}+100)^{2}}$$

$$ \frac {(20,000)^{3}} {(x_{1}+100)^{3}(x_{2}+100)^{3}(x_{3}+100)^{3}}$$

$$ \frac {(3,000)} {(x_{1}+100)(x_{2}+100)(x_{3}+100)}$$

**Q30 If X is the amount of money that a salesperson spends on gasoline during a day and Y is the corresponding amount of money for which he or she is reimbursed, the joint density of two random variables is given by $$f(x,y)=\frac{1}{25} \left ( \frac {20-x}{x} \right ); \; for \; 10 < x <20, \; x/2 <y <x $$, and $$f(x,y)= 0; \; elsewhere $$. Find the conditional density of Y given X = 12.**

$$h(y|12)= 1/6 \; for \; 6 < x < 12, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 2/3 \; for \; 1 < x < 2, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 1/3 \; for \; 1 < x < 6, \; h(y|12)= 0; \; elsewhere $$

$$h(y|12)= 2/9 \; for \; 0 < x < 12, \; h(y|12)= 0; \; elsewhere $$

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