Continuous Random Variables:

A continuous random variable can take on enough values to fill an entire interval.

It has a special curve associated with it called its probability density curve.

In general, a probability density curve must be on or above the horizontal axis, and the area of the region between the curve and the horizontal axis must be equal to 1.

The probability that a continuous random variable will take on a value between a and b is equal to the area under its probability density curve from a to b.

Examples:

1. Verify that the following curve can be a probability density curve for the random variable X that takes on values between 1 and 6.

X

Use the probability density curve to find the following probabilities:

If X represents measurements from a population, then what proportion of the measurements fall between 1 and 2?

2. Verify that the following curve can be a probability density curve for the random variable Y that takes on values between 1 and 5.

Y

Use the probability density curve to find the following probabilities:

If Y represents measurements from a population, then what proportion of the measurements fall between 3 and 5?

Normal Random Variables:

A continuous random variable whose probability density curve is a normal curve is called a normal random variable. A population whose measurements are represented by such a random variable is said to be normally distributed.

Normal probability density curves have a symmetric bell shape. The top of the bell is located at , the mean value of the random variable/population. The height of the top of the bell is determined by the value , the standard deviation of the random variable/population. The smaller is the higher the peak of the bell.

Normal random variables can take on any real number.

There is a special normal random variable with a mean of 0 and a standard deviation of 1. It’s called the standard normal, and it’s abbreviated as Z.

A table of cumulative areas under the standard normal curve has been compiled and is included near the end of the textbook starting on page T-1.

Find the following probabilities/areas:

Let’s reverse the process:

Find c so that

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12

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36

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1

15

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y

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1

2

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35

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4

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2

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1

16

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125

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142

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0

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172

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124

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48221

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13457

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15689

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1292

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8289

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