Question

An experiment results in either successes or failures each trial (denoted S and F ) and the total experiment is three trials. Here are the possible outcomes of the experiment
{SSS, SSF, SF S, F SS, SF F, F SF, F F S, F F F }
Let the random variable X denote whether or not a success occurred in the experiment. Assume each outcome above is equally likely.

a)    Fully define the probability distribution of X. Is this a probability mass function or a probability density function?
b) Fully define the cumulative distribution function of X.
c)    What is E(X)?
d)    What is σ2X?
e)    If the random variable Y represents the number of failures in the experiment, are X and Y indepen-dent? Why or why not?(3 pts)

Answer 1

3₃ SSS 0 - 2 1 0 1 2 2 - 118 1418 Give: possible outcomes for the three trials are Sss, SSF, SES, ESS ,SFE FSF FFS, FFF} preach outcome) =118 a let & denotes number of success occurred in this trial lakes values > 0,1,2,3; 0 FFF - SFF. FSF FFS, 2 SSF, SFS, FSS The probability mass function is given by Total PCD?o EP 1 PC*=) 318 318 is distribution is pmf. by the cummulative, probability distribution is given by Fru) 718 c> Ecx) = Ex. pran) = 0+1 +14.3 + 2x +3448 re Era) = 1.5 d) o C) = V(X) = (x2) -[Ex]? E(+2) = [² px) E (x²) = of tz +1+3 +4 +3+9+48 pax) 318 = 2418=3 VOX) = E(+2) - (ECO)? - 3 - 1.52 = 0.75 we know that & denotes number of success and given y denotes number of failure in the given experiment The joint pmf for tty is 13 7/8 PCX=0,450) = P(X=0). P(4-0) 3/8 marginal marginal 1 2 9/8 8 8 3 V8 0 4 - 318 o 1 2 O ve O 0 O O O joint D 318 - O 0 0 O 318 D O 18 o & 64 - 1/8 373 V8 from this we say that PC450,450) # PC*E0).poyo) That implies X and Y are not independant random variables