Given the discrete uniform population I(x)={ (1)/(3), x=2,4,6 0, . find the probabi

Given the discrete uniform population I(x)={ (1)/(3), ...

Key Concepts

Discrete Uniform Distribution

A discrete uniform distribution is one in which a finite number of outcomes are equally likely. In this type of distribution, each outcome has the same probability, making it straightforward to calculate expected values and probabilities. This concept is fundamental when dealing with randomly chosen numbers from a small set of possibilities with identical probabilities.

Population Mean and Variance

The population mean is the expected value of a random variable, calculated as a weighted average of all possible outcomes, while the variance measures the spread or dispersion of the outcomes around the mean. Determining these parameters is crucial for understanding the overall behavior of the population and is especially important when making inferences about sample means.

Sampling Distribution of the Sample Mean

When drawing a random sample from a population, the sample mean serves as an estimator for the population mean. The sampling distribution of the sample mean describes the probability distribution of these means over many samples. Understanding this distribution allows one to assess the variability and likelihood of various sample mean values, which is key for constructing confidence intervals and hypothesis tests.

Central Limit Theorem

The Central Limit Theorem (CLT) is a foundational statistical principle that states that, for a large enough sample size, the sampling distribution of the sample mean will be approximately normally distributed regardless of the original population's distribution. This theorem enables practitioners to apply methods based on the normal distribution to sample means even when the underlying distribution is not normal.

Standardization and Z-Scores

Standardization involves converting a random variable to a standard score, or Z-score, which measures how many standard deviations a value is from the mean. By using Z-scores, it becomes possible to utilize the standard normal distribution to calculate probabilities. This procedure is essential for comparing values from different normal distributions or for assessing probabilities within the standard normal framework.

Transcript

00:01 Right, in this question we are given the discrete uniform population where of x or x there is one -third when x is equal to 2, 4 or 6 or it is equal to 0 also.

00:18 So what we do right now? and the question says, find the probability that a random sample of size 54 selected with the replacement will yield a simple mean greater than 4.

00:31 One but less than 4 .4 assume the assume means to be measured to the nearest 10th so first of all we we are saying we get my red pen there our mean is right 54 and our meal there is for how do we get our meal or meal it is the expectation of x there which is just one -third plus by the addition of those figures of 0 of x which is 2 2 plus 4 plus 6 day and this will give us uh this is 6 plus 12 over 3 which is 4 so our mu is 4 there and then that means our variance variance of bikes of the sample the variance is variance over of the population there it will be uh sigma squared uh divided by n there our n is 54 but what is our variance our variance is the expectation uh squared there minus the expectation of x isn't it right so this is going to give us what that for this one is one third uh then one third of two squared right two squared okay two squared by one third four squared which is two squared plus squared plus four squared plus six squared isn't it it'll be minus it's a minus they're not an equal sign it will be minus expectation squared which is four squared which is sixteen this will be equals to five fifteen over three minus sixteen and this gives us 8 over 3 there.

02:38 So that's our virus.

02:40 There is 8 over 3.

02:41 Our n is 54.

02:42 You divide there.

02:43 You get 4 over 8 .1...

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