Exam 2 ECON E-370
binomial distribution
1. fixed number of Bermoulli trials (n) 2. each trial only has possibilities p and q (success and failure) 3. the probability of p and q are constant throughout the experiment 4. each trial is independent of the other trials in the experiment
For the continuous random variable, P(x ≤ x1 ) = ? a) P(x > x1 ) b) P(x < x1 ) c) P(x ≤ x1 ) d) P(x = x1 )
b) P(x < x1 )
In a standard normal distribution, the probability that z is greater than zero is: a. at least 0.5 b. 1 c. 0.5 d. 1.96
c. 0.5
BINOM.DIST()
(x, n, p, cumulative) ADJUST BOUNDS, not continuous!
cumulative probability
F(x) is used to calculate cumulative probability at any point or that is <= to any point. area under the curve f(x) to the left of the point.
probability density function (pdf)
area under the graph of f(x) between 2 points of interest. used to find the probability of a continuous random variable.
law of large numbers
as the number of trials or observation increases, the observed probability of an event (empirical probability) approaches theoretical (classical) probability
empirical method
assigning probabilities based on experimentation or historical data
subjective method
assigning probabilities based on judgment or experience
classical method
assigning probabilities based on the assumption of equally likely outcomes
A continuous random variable may assume: a. only integer values in an interval or collection of intervals b. any value in an interval or collection of intervals c. only fractional values in an interval or collection of intervals d. only the positive integer values in an interval
b. any value in an interval or collection of intervals
The center of a normal curve: a. is the standard deviation b. is the mean of the distribution c. is always equal to zero d. cannot be negative
b. is the mean of the distribution
types of discrete probability distributions
binomial
The normal distribution can be used as an approximation to the:
binomial distribution
For any continuous random variable, the probability that the random variable takes on exactly a specific value is: a. 0.5 b. 1 c. 0 d. any value between 0 to 1
c. 0
For the standard normal probability distribution, the area under the probability density function to the left of the mean is: a. any value between 0 to 1 b. Cannot say exactly without knowing the value of the mean c. 0.5 d. 1
c. 0.5
It is known that the length of a certain product x is normally distributed with µ = 20 inches. How is the probability P(x>16) related to P(x<16)? a. P(x>16) is smaller than P(x<16) b. No comparison can be made with the given information c. P(x>16) is greater than P(x<16) d. P(x>16) is the same as P(x<16)
c. P(x>16)P(x>16)is greater than P(x<16)P(x<16)
The normal distribution can well approximate the binomial distribution as long as: a. nq≥5 b. np≥5 c. np≥5,nq≥5 d. We do not need any additional requirements because it is always the case
c. np≥5,nq≥5
NORM.INV()
calculates the value of x that corresponds to a given value of the cumulative probability = NORM.INV(probability, mean, standard_dev) *Never do 1-NORM.INV()!!
NORM.S.INV()
calculates z value which corresponds to a given cumulative probability of the standard normal distribution NORM.S.INV(probability)
For a continuous random variable x, the probability density function f(x) represents: a. the area under the curve at x b. the probability at a given value of x c. the area under the curve to the right of x d. the height of the function at x
d. the height of the function at x
NORM.S.DIST()
function can also be used to find probabilities from the standard normal distribution when the z-score is KNOWN NORM.S.DIST(z, cumulative) where: cumulative = FALSE if you need the probability density function cumulative = FALSE (or 0) if you need the probability density function cumulative = TRUE (or 1) if you need the cumulative probability
NORM.DIST()
function is used to find normal probabilities NORM.DIST(x, mean, standard_dev, cumulative) cumulative = FALSE (or 0) if you need the probability density function cumulative = TRUE (or 1) if you need the cumulative probability
discrete probability distribution
listing of all the possible outcomes of an experiment for a discrete random variable, along with the relative frequency of each outcome. formula used to describe it is p(x), or a probability function
continuous random variable
may assume any numerical value in an interval or collection of intervals. often measured amounts, and fractions are possible (i.e. time, height, distance). because there are an infinite number of possible values, the probability of one specific value occurring is theoretically equal to 0. Instead, we determine the probability of an event falling within a given interval.
discrete random variable
may assume either a finite number of values or an infinite sequence of values. they are integers that are usually counted (i.e. number of TVs in a household)
what describes a normal distribution's shape?
mean (𝜇) and standard deviation (𝜎) completely describe its shape
expected value e(x)
mean of a discrete probability distribution, or the weighted average of all values of the random variable
normal dist can be used for binom dist when sample size is large enough so that:
np>=5 and nq>=5
binomial random variable
number of successes (x) in n trials
random variable
numerical description of the outcome of an experiment
mutually exclusive events
when one event occurs, the other event cannot occur
standard normal random variable
z
cumulative probabilities can be used to...
determine the probability of a random variable falling within a particular range of values
standard normal probability distribution
a random variable having a normal distribution with a mean of 0 and a standard deviation of 1
If x has a normal distribution with µ = 100 and σ = 5, then the probability P(90≤x≤95) can be expressed in terms of a standard normal variable z as: a. P(−2≤z≤−1) b. P(−2≤z≤−2) c. P(2≤z≤1) d. P(−2≤z≤1)
a. P(−2≤z≤−1)
Which of the following is NOT a characteristic of the normal probability distribution? a. The standard deviation must be 1 b. The distribution is symmetrical c. The mean of the distribution can be negative, zero, or positive d. The mean, median, and the mode are equal
a. The standard deviation must be 1
Which of the following can be represented by a continuous random variable? a. The time of a flight between Chicago and New York b. The number of defective light bulbs in a sample of five c. The number of arrivals to a drive-through bank window in a four-hour period d. The score of a randomly selected student on a five-question multiple-choice quiz
a. The time of a flight between Chicago and New York
A smaller standard deviation for the normal probability distribution results in a: a. fatter curve that is tighter and taller around the mean b. fatter curve that is more spread out around the mean and not as tall c. skinnier curve that is more spread out around the mean and not as tall d. skinnier curve that is tighter and taller around the mean
d. skinnier curve that is tighter and taller around the mean
normal probability distribution
1) bell shaped 2) symmetric 3) highest point is the mean which is also the median and the mode 4) the standard deviation determines the width of the curve: larger values result in wider and flatter curves 5) values near the mean, where the curve is the tallest, have a higher likelihood of occurring than values far from the mean, where the curve is shorter
are mutually exclusive events independent?
NO. They are dependent because if one event occurs the other cannot. Therefore, each event is reliant on the outcome of the other event's outcome.