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Continuous Probability Distributions
Recall that we have contrasted continuous variables from discrete variables by noting that while there are gaps between possible values of a discrete variable, a continuous variable can take any value in some interval of the number line.
Suppose X is a discrete random variable taking on the values 1 , 2 ,... , n with probability distribution function fX (x). Then
P [2 ≤ X ≤ 4] = fX (2) + fX (3) + fX (4)
Suppose X is a continuous random variable with probability density function fX (x). Then
P [a < X < b] =
∫ (^) b a f^ (x)dx^ (1)
Suppose X is a continuous random variable taking on any value x ∈ [c, d]. Then the probability density function satisfies
fX (x) > 0 if x ∈ [c, d] = 0 otherwise
∫ (^) d c fX^ (u)du^ = 1,
The probability P (a < X < b) equals ∫ (^) b a fX^ (u)du
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A random variable X is said to follow a Gaussian (Normal) distribution with population parameters μ and σ if its probability density function is given by
fX (x) = √ 21 πσ 2 e−^ (x^ −^ μ)
2 2 σ^2
A random variable X is said to follow the standard Normal distribution if its population parameters μ and σ equal 0 and 1, respectively. In this case,
fX (x) = √^12 π e−x
2 2
4 Example ( Normal distribution) Suppose X represents blood pressure and suppose that the population mean μ = 129 and the population standard deviation σ = 19.8. P [X > 150] =
∫ (^) ∞ 150
√^1
2 πσ^2 e−^ (x^ −^ μ)
2 2 σ^2 = ∫ (^) ∞ 150
√^1
2 π(19.8)^2 e−^ (x^ −^ (129))
2 2(19.8)^2
The probability that a Normal random variable takes on a value within an interval is equal to the area under the part of the normal density which lies above the interval.
Unfortunately, there is no simple formula for calculating this area, so we need to use a table.
Fortunately, we only need a table for the standard normal density with mean μ = 0 and standard deviation σ = 1.
If we want to calculate probabilities for a general normal random variable X with mean μ and SD σ, we need to construct a new random variable, called the standardized score of X, Z = X^ − σ μ.
Given values for μ and σ, we can actually go back and forth between the “X scale” and the “Z scale:” X = μ + σZ.
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Two simple rules can be very helpful in calculating normal probabilities:
- Since the total area under any density is 1, P [Z > z] = 1 − P [Z ≤ z].
- Since the normal density is symmetric about 0, P [Z < z] = P [Z > −z] = 1 − P [Z ≤ −z];
8 Suppose that X is a random variable that represents height. For the population of 18 to 74 year old women, height is normally distributed with mean μ = 63.9 inches and standard deviation σ = 2.6 inches. If we randomly select a woman from this population, what’s the probability that she is between 60 and 68 inches tall?
Statistical Inference:
The process of drawing conclusions about an entire population based on the information in a sample is known as statistical inference.
Q: I happen to “know” BP levels for all men in the U.S. follows a normal distribution with mean μ and standard deviation σ. You need to guess at μ and σ. How would you do this?
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Histograms of 100 sample means for different sample sizes (n).
16 Q: I happen to know that the number of car accidents in Madison each year follows a Poisson distribution. I know the mean (and so I know the variance). You need to guess at the mean. How would you do this?
Histograms of 100 sample means for different sample sizes (n).
Q: I happen to know that the number of successful surgeries out of 10,000 follows a Binomial distribution. I know the mean and variance. You need to guess at the mean and variance. How would you do this?
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Histograms of 100 sample means for different sample sizes (n).
20 The probability distribution of X¯ is called the sampling distribution of X¯. Understanding properties of the sampling distribution of X¯ allows us to make inference about population parameters based on a single sample! Characteristics that we observed in histograms which approximate the sampling distribution of X¯
- The mean of the sampling distribution is near the population mean from which the samples were taken (sample size n doesn’t matter).
- The variance of the sampling distribution gets smaller as the size of the sample (n) increases.
- For large sample sizes (n), the sampling distribution looks normal.
