Скачать или смотреть Continuous Probability Distributions: Normal, standard normal, calculate Probabilities CH7

1- Describe the characteristics of a normal probability distribution
2- Describe the standard normal probability distribution and use it to calculate probabilities

Continuous probability distributions are statistical distributions that represent the probability of a random variable taking on a continuous range of values. The two most commonly used continuous probability distributions are the normal distribution and the standard normal distribution.

The normal distribution is a continuous probability distribution that is symmetric, bell-shaped, and characterized by its mean (μ) and standard deviation (σ). The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1.

To calculate probabilities from the normal distribution, we use the area under the curve of the probability density function (PDF) of the distribution. This area represents the probability of a random variable taking on a certain range of values.

To calculate probabilities from the standard normal distribution, we use the Z-score, which represents the number of standard deviations that a particular value is from the mean. We can convert any normal distribution to a standard normal distribution by subtracting the mean from each value and then dividing by the standard deviation.

To calculate probabilities from the normal or standard normal distribution, we can use statistical software, statistical tables, or the standard normal distribution formula:

Z = (X - μ) / σ

where X is the random variable, μ is the mean, σ is the standard deviation, and Z is the corresponding Z-score.

Once we have the Z-score, we can use a standard normal distribution table to find the probability of a random variable is less than or greater than a certain value. We can also use software or calculators to find probabilities based on the normal or standard normal distribution.

For example, suppose the heights of adult men in a certain population are normally distributed with a mean of 70 inches and a standard deviation of 3 inches. What is the probability that a randomly selected man from this population has a height between 65 and 75 inches?

First, we standardize the lower and upper bounds:

Z1 = (65 - 70)/3 = -1.67
Z2 = (75 - 70)/3 = 1.67

We can then find the probability of a Z-score falling between -1.67 and 1.67 using a standard Normal table or software, which is approximately 0.90. Therefore, the probability that a randomly selected man from this population has a height between 65 and 75 inches is 0.90.

In summary, continuous probability distributions such as the Normal distribution and standard Normal distribution are important tools in probability theory and statistics. They allow us to model and calculate probabilities for continuous random variables, and their properties are widely used in statistical inference and hypothesis testing.

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#ZScoreExample
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#CumulativeDistributionFunctionExample
#PercentilesExample
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