Probstat/week5 practice 2

This is part of probstat.

Independent trials

1. There is an electric calculation circuit that, given the input, it gives the wrong answer with probability 0.1. What is the probability that we use the circuit for 10 times and always get the correct answers.

2. 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.

Note: the minimum value of K should be a function of the value of 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?