PROBABILITY AND STOCHASTIC PROCESSES

Probability and Stochastic Processes



PROBABILITY AND STOCHASTIC PROCESSES

Answer 1)

Coin toss probability is explored here with simulation. When asked the question, what is the probability of a coin toss coming up heads, most people answer without hesitation that it is 50%, 1/2, or 0.5 we get this probability by assuming that the coin is fair, or heads and tails are equally likely/ The probability for equally likely outcomes is:

Number of outcomes in the event ÷ Total number of possible outcomes

(a) Probability of winning Match in Australian Ground

Australia= 0.6

New Zealand= 0.4

Probability of Winning match in New Zealand

Australia= 0.4

New Zealand= 0.6

(i) The probability of New Zealand winning the series is (0.4) (0.6) (0.4) = 0.096

(ii) Given that New Zealand won the first game, the probability that they will go on to win the series is =

1- (0.4) (0.4) = 0.84

(iii) The probability that it takes three games to decide the series

= (0.40) (0.6) + (0.4) (0.6) + (0.6) (0.4)= 0.72

(b) The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome xi according to its probability, pi. The common symbol for the mean (also known as the expected value of X) is u. The law of large numbers states that the observed random mean from an increasingly large number of observations of a random variable will always approach the distribution mean. That is, as the number of observations increases, the mean of these observations will become closer and closer to the true mean of the random variable. This does not imply, however, that short term averages will reflect the mean.

(c) In probability theory, a probability density function, or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable's density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. The terms "probability distribution function"[1] and "probability function"[2] have also sometimes been used to denote the probability density function. However, this use is not standard among probability and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function".

Answer 2a)

(i) Proportions of people are heavier than 80 kg is approximately 8 people.

(ii)

1.The expected load of 18 people is 1224 kg

2.The Standard Deviation of the 18 people is ...