NOUN TMA Solutions: STT311 - Probability Distribution II

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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=3,4,6 and 7)=1/6 and f(5)=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 $$P(1 \leq X \leq 3)$$

0.369

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

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

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

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?

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

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 $$P(X=2, Y+Z \leq{3})$$

13/6 3

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

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

95/432

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

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

0.3

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## NOUN TMA Solutions: STT311 - Probability Distribution II

### Re: NOUN TMA Solutions: STT311 - Probability Distribution II

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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=3,4,6 and 7)=1/6 and f(5)=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 $$ g(X)=e^{3X/4}$$

4

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)

0.62

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

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?

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

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)

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

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 $$g(X,Y)= \frac {X}{Y^{3}} $$

15/84

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.

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

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.

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

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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=3,4,6 and 7)=1/6 and f(5)=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 $$ g(X)=e^{3X/4}$$

4

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)

0.62

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

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?

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

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)

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

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 $$g(X,Y)= \frac {X}{Y^{3}} $$

15/84

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.

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

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.

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

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### Re: NOUN TMA Solutions: STT311 - Probability Distribution II

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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=3,4,6 and 7)=1/6 and f(5)=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 $$ g(X)=e^{3X/4}$$

4

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)

0.62

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

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?

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

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)

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

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 $$g(X,Y)= \frac {X}{Y^{3}} $$

15/84

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.

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

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.

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

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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=3,4,6 and 7)=1/6 and f(5)=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 $$ g(X)=e^{3X/4}$$

4

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)

0.62

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

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?

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

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)

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

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 $$g(X,Y)= \frac {X}{Y^{3}} $$

15/84

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.

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

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.

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

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