STT311 TMA

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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