4. Distributions of Random Variables Flashcards
BINOMIAL DISTRIBUTION
What is the binomial distribution?
The binomial distribution is used to describe the number of successes in a fixed number of
trials.
The binomial distribution describes the probability of having exactly k successes in n independent Bernoulli trials with probability of a success p.
BINOMIAL DISTRIBUTION
What are the four conditions to check if it’s binomial?
(1) The trials are independent.
(2) The number of trials, n, is xed.
(3) Each trial outcome can be classied as a success or failure.
(4) The probability of a success, p, is the same for each trial.
BINOMIAL DISTRIBUTION
Formlar fyri:
The probability of observing exactly k successes in n independent trials.
The mean, variance, and standard deviation of the number of observed successes.
Suppose the probability of a single trial being a success is p. Then the probability of observing
exactly k successes in n independent trials is given by
(n) =p^k * (1 - p)^(n-k) = ( n! / (k! (n - k)! ) * p^k (1 - p)^(n-k)
(k)
The mean, variance, and standard deviation of the number of observed successes are
μ = np σ^2 = np(1 - p) σ = square root of( np(1 - p) )
NEGATIVE BINOMIAL
Wha is negative binomial?
The negative binomial distribution is more general: it describes the probability of observing
the kth success on the nth trial.
NEGATIVE BINOMIAL
What are the four conditions to check?
(1) The trials are independent.
(2) Each trial outcome can be classied as a success or failure.
(3) The probability of a success (p) is the same for each trial.
(4) The last trial must be a success.
BINOMIAL VERSUS NEGATIVE BINOMIAL
What’s the difference?
BINOMIAL VERSUS NEGATIVE BINOMIAL
In the binomial case, we typically have a xed number of trials and instead consider the number
of successes. In the negative binomial case, we examine how many trials it takes to observe a
xed number of successes and require that the last observation be a success.
NORMAL DISTRIBUTION
Among all the distributions we see in practice, one is overwhelmingly the most common. The
symmetric, unimodal, bell curve is ubiquitous throughout statistics. Indeed it is so common, that
people often know it as the normal curve or normal distribution.
Many variables are nearly normal, but none are exactly normal. Thus the normal distribution,
while not perfect for any single problem, is very useful for a variety of problems. We will use it
in data exploration and to solve important problems in statistics.
THE Z-SCORE
THE Z-SCORE
The Z-score of an observation is the number of standard deviations it falls above or below the
mean. We compute the Z-score for an observation x that follows a distribution with mean
and standard deviation using
Finding tail areas in R
The most common approach in practice is to use statistical software. For example, in the
program R, we could nd the area shown in Figure 4.6 using the following command, which
takes in the Z-score and returns the lower tail area:
…..> pnorm(1)
…..[1] 0.8413447
According to this calculation, the region shaded that is below 1300 represents the proportion
0.841 (84.1%) of SAT test takers who had Z-scores below Z = 1. More generally, we can also
specify the cuto explicitly if we also note the mean and standard deviation:
…..> pnorm(1300, mean = 1100, sd = 200)
…..[1] 0.8413447
WHat is a Bernoulli random variable?
When an individual trial only has two possible outcomes, often labeled as success or failure,
it is called a Bernoulli random variable.
If X is a random variable that takes value 1 with probability of success p and 0 with probability
1 - p, then X is a Bernoulli random variable with mean and standard deviation
mean = p
standard deviation = squareroot( p(1 - p) )
What is the geometric distribution?
The geometric distribution is used to describe how many trials it takes to observe a success.
NORMAL APPROXIMATION OF THE BINOMIAL DISTRIBUTION
The binomial distribution with probability of success p is nearly normal when the sample size
n is suffciently large that np and n(1 - p) are both at least 10.
IMPROVING THE NORMAL APPROXIMATION FOR THE BINOMIAL DISTRIBUTION
The normal approximation to the binomial distribution for intervals of values is usually improved
if cuto values are modied slightly. The cuto values for the lower end of a shaded region should
be reduced by 0.5, and the cuto value for the upper end should be increased by 0.5.
