Question

CHOOSE the correct answer and WRITE the corresponding letter in the provided space. 7. If X-Uniform(0.c), which of the following is CORRECT? A. E[X] = 0. B. P[X S c/2] = 1. C. Fx(c/2) = 0.5. D. fx(c/2) <fx©). 8. Assuming X is a continuous random variable, which of the following is INCORRECT? A x(x) = dFx(x)/dx. B. x(x) dx = 1. C fx(x) dx = Fzco). D. none of the above. 9. The CLT states that for Z = [X4, X ., Xn] where all X are iid, A. Z becomes Gaussian as n 00 B. Z becomes Gaussian as n 00 C. Z becomes Gaussian as n → D. none of the above. 10. Suppose that the GPA average of students is 3.0. Assuming a Gaussian distribution, what is the probability that a student selected at random has a GPA less than 3.0? A. 0.5 B. 0 C. 1 D. cannot be determined 11. Let X be a continuous random variable with sample space Sy = (x,x). Which of the following is INCORRECT? A. Fx(-o) = 0 B. Fx(x) s Fx(x2) c fx(x) > 0 if x <x< x2 D. fx(x) < 0 if x < x; 12. Assuming A-Gaussian(18, V12) and B-Uniform(12,24), which of the following is INCORRECT? A. E[A] = E[B] B. Var[A] + Var[B]. CPIAS 0] = F (0) D. P[B S 0] =0 13. Which continuous random variable is equivalent to a binomial discrete random variable? A. Gaussian random variable B. Uniform random variable C. Exponential random variable D. none of the above

Answer 1

0 < X < sumes x Writom (0,c) so f(x) = < .0**<C So, EGX) = (to P[KS ] - $**dx = { [*],? A now, fxzę ] = f[x55]: 0.5 So, flowest answer vis [G]:fx [C%22=05, the lower celseferde te [drouwen in D : vene of the above This is because all the following statements are correct : • fx(x) = . Fx (x) '» FxCat) = J. *(x) dx. [v] Leon BCH) = 1 => sleea, under curve is always 1 [v] .f(x) = fxcoo) ; now Fx Cao) = lim frica) = 1, i s of(x) = fx600) = 1 [u]. idustuve us Bi Zzi becomes Gaussion as n ol. The CLT describes the behavior of sums of randam variables for - sufficiently lange sample size , i.e. fox n-o, the alistribution o serve of raudiem variables is approximately Gaussian : Let x denate GPA a students such that E(X)= 3 & Sloandard deviation of x un socx) = 8x so, 2 <0] P[ X<3] = P[ Xo3 < , Z=323, NNCO, 5 csjstanf-1:33 = P[ 2<0]=0.5 Scanned with CamScanner [: Standard normal Curve is symmetric * about Mean o ].

er i 24 +12 2 12 12 ausilo answer is D. fx(x) <o it all This is because pdf u never negative i.e. O= f(x) < I always. If Quesja, A v Gaussion 18, Jiz) and B » UC12,24) ·mow, ECA) = 18 , 6CB) = 24+12 = 36 =18 So, ECA) = ECB ) · Vax (B) = (24-12]? = 144 = 12, But Vou (A) = (272)2 =12 So, vou CA) = Vou (B). • By def", Fx (x ) = P[X< x] = FALO) = P(A30] .:P[650] = o suice range of B is 12 to 24. So, Answer is Bi Vau [] # van CB ) | Quest, Answer is D: none of the above No continuous Randem variabele is equivalent to Benonial con Discrete Randens variable.' Scanned with CamScanner