Probability and Stochastic Processes

In a fair coin toss, there are two equally likely outcomes: heads or tails. Therefore, the probability of getting heads is 1/2.

The expected value of a random variable X is defined as E(X) = ΣxP(X = x). In this case, E(X) = 1(1/3) + 2(1/3) + 3(1/3) = 2.

The probability of A given B is P(A | B) = P(A ∩ B) / P(B). Therefore, P(A | B) = 0.2 / 0.6 = 0.33.

A continuous random variable can take on any value within a specified range. The height of a randomly selected person is an example of a continuous random variable because it can take on any value between a minimum and maximum height.

The probability density function of a standard normal distribution is given by $f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$.

The expected number of arrivals in a Poisson process with rate λ over a time interval of length t is given by λt.

The probability of transitioning from state S1 to state S3 in two steps is given by the product of the transition probabilities P(S1, S2) and P(S2, S3).

A stochastic process is a collection of random variables indexed by time. Stock prices over time are an example of a stochastic process because they are a collection of random variables (the stock prices) indexed by time.

The probability of getting at least one head in two coin tosses is the complement of the probability of getting no heads. The probability of getting no heads is (1/2)^2 = 1/4. Therefore, the probability of getting at least one head is 1 - 1/4 = 3/4.

The expected value of a random variable X with probability density function f(x) is defined as E(X) = ∫xf(x)dx. In this case, E(X) = ∫x\frac{1}{2}e^{-|x|}dx = 1.

The probability of both A and B occurring is given by P(A ∩ B) = P(A)P(B). Therefore, P(A ∩ B) = 0.4 * 0.6 = 0.24.

A discrete random variable can take on only a countable number of values. The number of heads in 10 coin tosses is an example of a discrete random variable because it can take on only the values 0, 1, 2, ..., 10.

The probability of getting exactly two heads in three coin tosses is given by the binomial distribution with n = 3 and p = 1/2. The probability of getting exactly k heads in n coin tosses is given by P(X = k) = (n choose k)p^k(1-p)^(n-k). In this case, P(X = 2) = (3 choose 2)(1/2)^2(1/2)^1 = 3/8.

The expected value of a random variable X is defined as E(X) = ΣxP(X = x). In this case, E(X) = 1(1/6) + 2(1/6) + 3(1/6) + 4(1/6) + 5(1/6) + 6(1/6) = 3.5.

There are 36 possible outcomes when rolling two fair dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36 = 1/6.