Question

1.Roll 3 times independently a fair dice. Let X = The # of 6's obtained. The possible values of the discrete random variable X are:

2.For the above random variable X we have E​[X] is:

3.The Domain of the moment generating function of the above random variable X is:

4.Let M(t) be the moment generating function of the above random variable X. The M'(0) is:

5.A discrete random variable X has the pmf f(x)=c(1/8)^x, for x in{8, 9, 10, ...}. Then c is:

6.Let X be a random variable such that Var(2X+3)=cVar(X). Then c is:

7.If the possible values of the discrete random variable X are 2, 4, 6, taken with the respective probabilities 1/2, 1/3, 1/6, then M"(0) is the following (M is the moment generating function of X):

8.The continuous random variable X has the density f(x)=c(x+10)^-2 on ​[0,40], and 0 otherwise. Then c is:

9.The probability that the above random variable X takes values less than 5 is:

Answer 1

ne the valuar .. 1 A fair die is rolled 3 times independently, X : # of os obtained. Since, the die is fair, and it is rolled 3 timer the outcome 6 may be obtained once, twice, thrice or maybe never. Therefore, the zv. X can take valuep: 0,1,2,3 To calculate E(X), Let us first find P[x=x] P[x=0] = No 6's in 3 tosses = ( 5 )3 P[X=1] = Only one 6 is 3 tossep = 13)(1)(52 P[x=2] = two 6°8 in 3 tosses = ZEE) P[x=3] = three c'e in 3 tossex = 3 -: E(X)= O x P[X=0] +1. P[x=1] +2.P[X=2] +3. P[x=3] = 0 +1.3.5+ 2.3,5 + 3. To = 108 = 10.5] 3) The moment generating function is defined by - Mx (t) = Eletx) 9) Mý (t) lt-e = E(xetico) = E(x) ....M (0) =E(X) = 0.5 5) f(1) = c(4), 2648,9,10,...} We know that {f(x) = 1 216 p= 8 + c(4) - (1) 8 8 = 1 (C= 7.87 |

6 nes thric x bea no such that, Varo (2x+3)= c Var (x) We know that, variance is invariant of scale change. ... Var (2x+3)= var(2x) = y 22 Vor(x) = 4 Varex) Hence [c=4] The RV x takes the values 2,4,6 with respective probabilies 1/2, 1/3, 46. Mx (t) = Eletx) Mx (t) = E (xetx) MX(t) = E(x2 etx) Mý (t)| + -o = M" (O) = E(x2) .. E(x2) = 3x2.P[x=x] 22. 2 / 2 + 4². I + 6. 1 / 2 = 2+ +6 = 24446 - 40 =13.33 M"CO) – 13.33]