Probstat/week5 practice 2

This is part of probstat.

Independent trials

1. There is a random experiment whose outcomes are successful with probability 1/2. We would like to perform the experiment for K times. What is the probability that none of the outcomes from these K experiments is successful. What is the minimum value of K if we want the probability to be less than 1/n², for some value n.

Independent random variables

1. Consider a simple random experiment of choosing, uniformly at random, an integer from the set {0,1}.

• 1.1 Let X be a random variable that denotes the chosen integer from the experiment. Find E[X] and Var(X).
• 1.2 Let's perform the experiment 2 times. Let X₁ and X₂ be random variables that denote the chosen integers from the first and the second experiments. Let Y = (X₁ + X₂)/2. Find E[Y] and Var(Y).
• 1.3 Let's perform the experiment K times. Let X₁, X₂,... X_K be random variables that denote the chosen integers from each experiments. Let Z = (X₁ + ... + X_K)/K. Find E[Z] and Var(Z).
• 1.4 Compare these 3 random variables. I.e., compare E[X], E[Y], and E[Z], and also Var(X), Var(Y), and Var(Z). What is the effect of the number independent trials to the variance. What does it actually mean?