If random variable X takes on the values of -1,0 and 2 with probabilities 0.3, 0.3, and 0.4 respectively, what is its expected value, E(X)?

A particle executes an unrestricted simple random walk on the line with p=0.7 and q=0.3. Its position at time n is represented by the random variable Xn, n = 0, 1,...; Xo = 0. (a) Calculate the values of the following probabilities. (i) P(X5 = -2) (ii) P(X6 = 2) (b) Calculate the probability that the particle returns to its starting point for the first time after 8 steps. (c) After the particle has been moving for some time, it is observed to be located at the point +3. (i) Calculate the probability that the particle will ever return to the origin from the point +3. (ii) Calculate the probability that starting from the point +3, the particle will visit the point +8 without first visiting the origin.

In a simulation study on cyber attacks, the researcher is interested to investigate on whether the attack(s) are successfully performed. Let A define the event that an attacker has successfully performed the attack. If Pr(A) = 0.13, what is the probability Pr(A')?