In the 1910 article "The Probability Variations in the Distribution of αParticles" (Philosophica

step 1 Hello everyone, this is problem 85. Y is equal to the number of alpha particles reaching a scree

In the 1910 article "The Probability Variations in the...

In the 1910 article "The Probability Variations in the Distribution of $\alpha$ Particles" (Philosophical Magazine, Series 6, No. 20, pp. 698-707), E. Rutherford and H. Geiger described the results of experiments with polonium. The experiments indicate that the number of $\alpha$ (alpha) particles that reach a small screen during an 8 -minute interval has a Poisson distribution with parameter $\lambda=3.87 .$ Determine the probability that, during an 8 -minute interval, the number, $Y,$ of $\alpha$ particles that reach the screen is a. exactly four. b. at most one. c. between two and five, inclusive. d. Construct a table of probabilities for the random variable $Y$. Compute the probabilities until they are zero to three decimal places. e. Draw a histogram of the probabilities in part (d). f. On average, how many alpha particles reach the screen during an 8-minute interval?

In the 1910 article "The Probability Variations in the Distribution of $\alpha$ Particles" (Philosophical Magazine, Series 6, No. 20, pp. 698-707), E. Rutherford and H. Geiger described the results of experiments with polonium. The experiments indicate that the number of $\alpha$ (alpha) particles that reach a small screen during an 8 -minute interval has a Poisson distribution with parameter $\lambda=3.87 .$ Determine the probability that, during an 8 -minute interval, the number, $Y,$ of $\alpha$ particles that reach the screen is
a. exactly four.
b. at most one.
c. between two and five, inclusive.
d. Construct a table of probabilities for the random variable $Y$. Compute the probabilities until they are zero to three decimal places.
e. Draw a histogram of the probabilities in part (d).
f. On average, how many alpha particles reach the screen during an 8-minute interval?

Key Concepts

Expected Value of a Poisson Distribution

The expected value of a Poisson distribution is equal to its parameter ?. This value represents the average number of events occurring within the defined interval and provides a measure of the central tendency of the distribution, serving as a key summary statistic in probability and statistics.

Histogram Representation of a Discrete Distribution

A histogram is a graphical representation that uses bars to show the frequency or probability of outcomes for a discrete random variable. Each bar's height corresponds to the probability from the PMF, helping visualize the distribution of the variable, understand its shape, and identify which outcomes are most or least likely.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event. It is commonly used in scenarios where events occur randomly and independently, such as counting occurrences of rare events over time.

Probability Mass Function (PMF) of the Poisson Distribution

The PMF of a Poisson random variable gives the probability of observing exactly a certain number of events in the fixed interval. It is defined as P(Y = y) = (e^(-?) * ?^y) / y!, where ? is the average rate, y is the number of occurrences, and e is Euler's number. This formula is central in calculating specific probabilities for given event counts.

Cumulative Distribution Function (CDF) for Discrete Distributions

The CDF for a discrete distribution, such as the Poisson distribution, represents the probability that the random variable takes on a value less than or equal to a particular number. It is obtained by summing the PMFs for all values up to that number, which is especially useful when determining 'at most' or 'between' type probabilities.

Transcript

00:01 Hello everyone, this is problem 85.

00:06 Y is equal to the number of alpha particles reaching a screen.

00:34 Now, y is distributed as a parson with lambda equal to 3 .87, which is our mean.

00:51 And little y can take on values 0, 1, and so on.

01:01 The probability that capital y is equal to little y is equal to lambda to the y, e to the negative lambda over y factorial.

01:18 For part a, we want to know what is the probability that 1, y is equal to 4 which is equal to 3 .87 to the 4 e to the negative 3 .87 over 4 factorial we will use excel to figure it out now for part b we have the probability that y is less than or equal 1 meaning why could only take on the values you're in 1 so it's equal to probability of y is equal to 0 plus the probability the y is equal to 1.

02:11 And for part c we want to find the probability that y takes on values between 2 and 5 inclusive meaning that it could take on values 2 three, four, and five, the y.

02:35 Okay, now for part d and e, part d, we just need to figure out some probabilities, and part e draw a histogram.

02:51 Okay, so if we want to do this in excel, you need to use this part right here.

03:00 So you just write down the equal sign because that's how functions are taken in excel and you type pusson .dd so we know we're talking about the person distribution.

03:14 The zero there is because i'm trying to find the probability that y is equal to zero and the 3 .87 that's your lambda right here which is the mean and you also need to put false because we're not talking about the cumulative distribution...

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