So the formula for um this is gonna be binomial, so it's gonna be 10 c four. 4 and we want X two before because that's the number of females are interested. When we talk about a coin toss, we think of it as unbiased: with probability one-half it comes up heads. So in this case we have and it's 10 P is probably female sets. And we want um To find the probability that a class of 10 students will have exactly a certain number of females for females. The second part particular college, 60% of the students are female. How many coin ips on average does it take to get n consecutive heads The process of ipping n consecutive heads can be described by a Markov chain in which the states correspond to the number of consecutive heads in a row, as depicted below. And if we take the screws of three we got 1.73, so that's our variance. And then the standard deviation which is just the square root of the variance, so that's just gonna be a square root three.
Uh And then the variants, which is sigma squared, S squared I guess since we have a sample, so s squared, it's going to be n times p times Q Q being one minus P. Um I'm going to calculate the mean variance and standard deviation of the number of heads of the mean X bar is just N times P, Which is 12 tosses times. To better understand this issue, let's look at an example.Alright, so we have a fair coin flip 12 times, we have to count the number of heads. For a binomial distribution, the parameters are n, p, and q. Flipping coins comes under the binomial distribution. In some sense, both are implying that the number of arrivals in non-overlapping intervals are independent. Assuming that the coins are unbiased, the answer is 2.5. This may not sound so important, but it becomes important when we combine it with the follow-ing rule: This equation can be derived directly from the expectation formulas, and is highly intuitive. Thinking of the Poisson process, the memoryless property of the interarrival times is consistent with the independent increment property of the Poisson distribution. If we ip just one coin, we get a 0 half the time and a 1 half the time, for a variance of 0.25. Remember that a discrete random variable $X$ is said to be a Poisson random variable with parameter $\mu$, shown as $X \sim Poisson(\mu)$, if its range is $R_X=\a,x \geq 0. Poisson random variable: Here, we briefly review some properties of the Poisson random variable that we have discussed in the previous chapters. In practice, the Poisson process or its extensions have been used to model $-$ the number of car accidents at a site or in an area $-$ the location of users in a wireless network $-$ the requests for individual documents on a web server $-$ the outbreak of wars $-$ photons landing on a photodiode. Thus, we conclude that the Poisson process might be a good model for earthquakes. Other than this information, the timings of earthquakes seem to be completely random. For example, suppose that from historical data, we know that earthquakes occur in a certain area with a rate of $2$ per month. It is usually used in scenarios where we are counting the occurrences of certain events that appear to happen at a certain rate, but completely at random (without a certain structure). The Poisson process is one of the most widely-used counting processes.