## NOUN TMA Solutions: STT311 - Probability Distribution II

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

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}$$serve as the values of the probability distribution of a random with countably infinite range x=0,1,2,􀳦?􀳦􀳦?􀳦􀳦?􀳦􀳦?􀳦􀳦?􀳦..? no value Q10 The probability distribution of V, the weekly number of mangoes that will freely from the tree at a certain region is given by g(0) = 0.40, g(1) = 0.30, g(2) = 0.20 and g(3) = 0.10 , find the probability that there will be at least 2 mangoes that will freely in any one week 0.3 Solutions2tma@gmail.com Whatsapp: 08155572788 Stay informed when you download our app https://play.google.com/store/apps/deta ... m.nounites Jed Posts: 892 Joined: Tue Oct 10, 2017 6:37 pm Contact: ### Re: NOUN TMA Solutions: STT311 - Probability Distribution II Solutions2tma@gmail.com Whatsapp: 08155572788 Stay informed when you download our app https://play.google.com/store/apps/deta ... m.nounites Q11 An urn contains four balls numbered 1, 2,3 and 4. If two balls are drawn from the urn at random and Z is the sum of the numbers on the two balls drawn, find the probability distribution of Z. f(x=3,4,6 and 7)=1/6 and f(5)=1/3 Q12 If X has the probability density$$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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Jed
Posts: 892
Joined: Tue Oct 10, 2017 6:37 pm
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### Re: NOUN TMA Solutions: STT311 - Probability Distribution II

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Whatsapp: 08155572788

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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