6.1 The Standard Normal Distribution 1. Fill in the blanks using both our textbook and Lumen textbook. 2. A normal distribution is a continuous probability density function used to model the probability distribution of a random variable. The functions of all normal distributions are already defined: f(x) = So, every normal distribution is completely determined by the two parameters . and ., which are called the mean and the standard deviation. 3. If X ~ N(3, 1), then P(X = 4) = 0 and P(X < 4) = P(X < 4) because every normal distribution function is a probability density function of a continuous random variable. 4. When X ~ N(µ,0), a probability P(a < X < b) is given as the area between the graph and the x-axis over [a,b]. 5. One way of using normal distribution modeling: For a random variable whose mean and standard deviation are known, we assume that the random variable fits the normal distribution with the same mean and standard deviation. Then, we use the normal distribution to find the probabilities of the variable. 6. Notation. X ~ N(µ,0) means that

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!