Suppose X has a Poisson distribution with a mean of 4 . Determine the following probabilities: (

Suppose X has a Poisson distribution with a mean of 4 ....

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Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, with the assumption that these events occur with a known constant mean rate and independently of the time since the last event. It is widely used for modeling count data and rare events.

Probability Mass Function

The probability mass function (PMF) of a Poisson distribution is given by the formula P(X = k) = (e^(??) * ?^k) / k! for k = 0, 1, 2, …, where ? is the average rate (mean) of occurrence. This formula is fundamental for computing the probability of exactly k events occurring in the specified interval.

Cumulative Distribution Function

The cumulative distribution function (CDF) for a Poisson random variable is the sum of the probabilities obtained from the PMF for all values up to and including a specified value. This is used to compute probabilities such as P(X ? k) by summing the individual probabilities from k = 0 to the desired value.

Transcript

00:02 In this problem it is given that x has a poison distribution with mean 4.

00:09 That is mean mu is equal to 4.

00:14 We know that mu is equal to lemra t.

00:18 So lemra t is equal to 4.

00:23 For poison variable x, we have probability of x is equal to x is e raised to minus lemra t.

00:32 Lambda t raised to x divided by x factorial where x goes from 0 1 up to infinity so probability of x is equal to x is e raised 2 minus we have lambda t is equal to 4 so e raised 2 minus 4 into 4 into 4 raised to x divided by x factorial here first we have to find probability of x is equal to 0.

01:17 Probability of x is equal to 0.

01:24 This is equal to e raised to minus 4.

01:31 4 raised to 0 as here x is equal to 0.

01:37 So 4 raised to 0 divided by 0 factorial.

01:44 This is equal to 0.

01:45 This is equal to e raised to minus 4, 4 raised to 0 is 1 and 0 factorial is also 1.

01:54 So, e raised 2 minus 4, which is equal to 0 .0183.

02:07 So, probability of x is equal to 0 is 0 .0183.

02:14 Next, we have to find probability of x less than or equal to 2.

02:25 Probability of x less than or equal to 2.

02:31 This is equal to the value of x starts from 0.

02:40 So probability of x is equal to 0 plus probability of x is equal to 1 plus probability of x is equal to 1 of x is equal to 2...