Chapter 7 - QMB2100 Flashcards
What are the 3 continuous distribution probabilities studied?
- Uniform probability distribution.
- Normal probability distribution.
- Exponential probability distribution.
When do you use the uniform probability distribution?
When we do not have any information regarding the shape of a random variable’s probability distribution. When we have no information that an event is more likely than any other, then every event is equally likely.
What do you need to establish a uniform probability distribution?
The minimum and maximum values.
How are uniform probability distributions represented?
Rectangular shaped determined by a (minimum value), b (maximum value) and height (1/b-a).
What is the formula for P(x) in a uniform probability distribution?
P(x) = 1/(b - a)
What is the mean for a uniform probability distribution?
μ = (a + b) / 2
What is the standard deviation for a uniform probability distribution?
σ = sqrt[(b-a)^2 / 12)]
Describe the area of a uniform probability distribution.
Area = height * base = 1.00; 1 / (b - a) * (b - a) = 1.00
What are the characteristics of a normal probability distribution?
- It is bell shaped and has a single peak at the center of the distribution. The mean, median, and mode are equal. The total area under the curve is 1.00.
- It is symmetrical about the mean.
- It is asymptotic on the X-axis.
- The location is determined by the mean and the dispersion is determined by the standard deviation.
What is the formula for the normal probability distribution?
P(x) = {[1/(σ * sqrt(2π))] * e} - [(x - μ)^2 / 2σ^2]; where σ standard deviation, μ is the mean, x is the random continuous variable, π is 3.1416, and e is 2.718.
What is the standard normal distribution?
A normal distribution with a mean equal to 0 and a variance equal to 1.
How do you convert a normal probability distribution to a standard normal probability distribution?
By subtracting the mean from each observation and dividing this difference by the standard deviation. The results are called z values.
What is the formula for the standard normal value?
z = (x - μ) / σ; where x is the random variable, μ is the mean, and σ is the standard deviation.
How do you find the area between 0 and z or -z?
Look up the probability table directly.
How do you find the area beyond z or -z?
Locate the probability of z in the table and subtract it from 0.5000.
How do you find the area between two z points on different sides of the mean?
Determine the z values and add the corresponding probabilities.
How do you find the area between two z points on the same side of the mean?
Determine the z values and subtract the smaller one from the larger one.
How do you calculate the value of the random variable x?
Using the z value formula from the standard normal value:
z = (x - μ) / σ
x = μ + zσ
What are the implications of the empirical rule?
- 68% of the observations lie within +/- 1 standard deviations of the mean.
- 95% of the observations lie within +/- 2 standard deviations of the mean.
- 99.7% of the observations lie within +/- 3 standard deviations.
What are 5 characteristics of the exponential probability distribution?
- Usually describes the time between events.
- The actions occur independently at a constant rate per unit of time or length.
- The exponential random variable x is always positive.
- The exponential distribution is always positively skewed.
- The distributions is described by only one parameter λ referred to as the rate parameter.
What is the mean of the exponential probability distribution?
mean = 1/λ
What is the standard deviation of the exponential probability distribution?
standard deviation = 1/λ
What is the formula for the exponential distribution?
P(x) = λe^(-λx)
How do you find the probability using the exponential distribution?
P(arrival time < x) = 1 - e^(-λx)
