Probstat/week5 practice 1

Practice exercises

Variances

1. Consider a random variable Z that becomes 1 with probability 1/3, 0 with probability 1/3, and 1 with probability 1/3. Find E[Z], and Var(Z).

2. Let random variable X be the outcome after tossing a fair die (with 6 faces of values 1,2,3,...,6). Find E[X] and Var(X).

3. You have observed the distribution of the score for a probability class. You notice that, from an exercise, students get 5 points with probability 0.1, 4 points with probability 0.2, 3 points with probability 0.3, 2 points with probability 0.1, 1 point with probability 0.2, and 0 points with probability 0.1. Let random variable X be the points a student will get. Find E[X] and Var(X).

Binomial and geometric random variables

1. Your friend claims to have a fair coin. You would like to prove it so you take her coin and toss it for 20 times. You got 15 heads. Suppose that the coin is truely a fair coin, what is the probability that this happens. Now, given that result, do you believe that the coin is indeed a fair one?

Theoretical exercises

1. For a random variable X with variance Var(X). For a constant a, prove that $Var(aX)=a^{2}Var(X)$.

2. For a non-negative discrete random variable $X$ that takes on values 0,1,2,..., show that $\mathrm {E} [X]=\sum _{i=0}^{\infty }P\{X\geq i\}$.