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1. Describe the following and give examples of each.
a) Discrete random variable
A discrete random variable is one which may take on only a countable number of distinct
values such as 0, 1, 2, 3, 4, ... Discrete random variables are usually (but not necessarily)
counts. If a random variable can take only a finite number of distinct values, then it must be
discrete.
Examples of discrete random variables include the number of children in a family, the Friday
night attendance at a cinema, the number of patients in a doctor's surgery, the number of
defective light bulbs in a box of ten.
b) Continuous random variable
A continuous random variable is one which takes an infinite number of possible values.
Continuous random variables are usually measurements.
Examples include height, weight, the amount of sugar in an orange, the time required to run
a mile.
c) Probability
probability provides a quantitative description of the likely occurrence of a particular event.
Probability is conventionally expressed on a scale from 0 to 1; a rare event has a probability
close to 0, a very common event has a probability close to 1.
The probability of an event has been defined as its long-run relative frequency. It has also
been thought of as a personal degree of belief that a particular event will occur (subjective
probability).
In some experiments, all outcomes are equally likely. For example if you were to choose one
winner in a raffle from a hat, all raffle ticket holders are equally likely to win, that is, they
have the same probability of their ticket being chosen. This is the equally-likely outcomes
model and is defined to be:
Examples
1. The probability of drawing a spade from a pack of 52 well-shuffled playing cards is:
since:
Event E = 'a spade is drawn'
The number of outcomes corresponding to E = 13 (spades) and
The total number of outcomes = 52 (cards)
2. When tossing a coin, we assume that the results 'heads' or 'tails' each have equal
probabilities of 0.5.
d) Binomial experiment
Definition of Binomial Experiment
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A binomial experiment is an experiment with a fixed number of independent
trials.
In a binomial experiment each trial has exactly two outcomes.
The probability of each outcome in a binomial experiment remains the same
for each trial.
Examples of Binomial Experiment
Flip a coin 30 times to see how many heads you get.
Here,
1. the number of trials is fixed, 30, and each trial is independent of the
other
2. the outcome can only be a head or a tail in each trial
3. the probability of getting a head is the same as the probability of
getting a tail. So, its a binomial experiment.
e) Population parameter
population parameter is a way of summarizing a probability distribution, and a sample
statistic is a way of summarizing a sample of observations.
Examples include mean and variance. A population parameter summarizes the probability
distribution from which it is assumed a sample has been drawn. A sample statistic summarizes
the sample itself.
For most purposes, we want to define a sample statistic so that it is an unbiased estimator of
the corresponding population parameter. To be unbiased, the expected value of the sample
statistic (under the assumed probability distribution) must equal the corresponding population
parameter. In addition, among the possible unbiased estimators, we usually want to choose a
statistic with low, if not the least, variance around the population parameter.
From example, if m is the population mean and m is our defined sample mean, we want a
definition of m such that:
E(m) = m and E[(m - m)²] is small
f) Probability distribution
The probability distribution of a discrete random variable is a list of probabilities associated
with each of its possible values. It is also sometimes called the probability function or the
probability mass function.
More formally, the probability distribution of a discrete random variable X is a function which
gives the probability p(xi) that the random variable equals xi, for each value xi:
p(xi) = P(X=xi)
It satisfies the following conditions:
a.
b.
g) Standard score
A Standard Score indicates how far a particular score is from a test's average. The unit that tells the
distance from the average is the standard deviation (sd) for that test. For the PPVT-III and the WAIS III, the average is 100 and the sd is 15. The standard deviation (sd) is always given for a standard
score. Standard Scores between -1 sd (85) and +1 sd (115) fall in the normal range on the ability being
tested.
Above + 1 sd (115+) a learner is in the top 15% of performances. Below -1 sd (-85) , she/he is in the
lowest 15% of performances
h) central limit theorem
The Central Limit Theorem states that whenever a random sample of size n is taken from any
distribution with mean µ and variance
, then the sample mean
will be approximately
normally distributed with mean µ and variance
/n.
The larger the value of the sample size n, the better the approximation to the normal.This is
very useful when it comes to inference.
For example, it allows us (if the sample size is fairly large) to use hypothesis tests which
assume normality even if our data appear non-normal. This is because the tests use the
sample mean
distributed.
, which the Central Limit Theorem tells us will be approximately normally
i) Standard error of the mean
The standard error of a method of measurement or estimation is the estimated standard
deviation of the error in that method. Namely, it is the standard deviation of the difference
between the measured or estimated values and the true values.
j) Normal distribution
probability distribution shaped like a bell, often found in statistical samples. The distribution of
the curve implies that for a large population of independent random numbers, the majority of
the population often cluster near a central value, and the frequency of higher and lower values
taper off smoothly.
k) Standard normal distribution
The normal distribution with mean zero and variance 1 is referred to as the standard
normal distribution. This is often denoted by Z ~ N(0,1) and has the probability density
function
The standard normal distribution is sometimes called the z distribution. A z score always
reflects the number of standard deviations above or below the mean a particular score is. For
instance, if a person scored a 70 on a test with a mean of 50 and a standard deviation of 10,
then they scored 2 standard deviations above the mean. Converting the test scores to z
scores, an X of 70 would be:
So, a z score of 2 means the original score was 2 standard deviations above the mean. Note
that the z distribution will only be a normal distribution if the original distribution (X) is
normal.
l) Continuity correction factor
Because the normal distribution can take all real numbers (is continuous) but the binomial
distribution can only take integer values (is discrete), a normal approximation to the binomial
should identify the binomial event "8" with the normal interval "(7.5, 8.5)" (and similarly for
other integer values). The figure below shows that for P(X > 7) we want the magenta region
which starts at 7.5.
Example: If n=20 and p=.25, what is the probability that X is greater than or equal to 8?
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The normal approximation without the continuity correction factor yields
z=(8-20 × .25)/(20 × .25 × .75)^.5 = 1.55, hence P(X *greater than or equal to* 8)
is approximately .0606 (from the table).
The continuity correction factor requires us to use 7.5 in order to include 8 since the
inequality is weak and we want the region to the right. z = (7.5 - 5)/(20 × .25 ×
.75)^.5 = 1.29, hence the area under the normal curve (magenta in the figure above)
is .0985.
The exact solution is .1019 approximation
Hence for small n, the continuity correction factor gives a much better answer
2. Explain the difference between a discrete and a continuous random variable. Give two
examples of each type or random variable.
Answer:
If a variable can take on any value between two specified values, it is called a continuous
variable; otherwise, it is called a discrete variable.
Examples:

Suppose the fire department mandates that all fire fighters must weigh between 150
and 250 pounds. The weight of a fire fighter would be an example of a continuous
variable; since a fire fighter's weight could take on any value between 150 and 250
pounds.

Suppose we flip a coin and count the number of heads. The number of heads could be
any integer value between 0 and plus infinity. However, it could not be any number
between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the
number of heads must be a discrete variable.
3. Determine whether each of the distributions given below represents a probability
distribution. Justify answer
Solution:
The sum of probabilities must be equal to 1.
a) x |1 | 2 | 3 | 4
P(x) |1/8| 1/8| 3/8 | 1/8
Solution:
Here the sum is
6
 1 , so isn't a probability distribution
8
b) x | 20 | 30| 40 | 50
P(x)| 0.3|0.2| 0.1| 0.4
Solution:
Here sum is equal to 1, so it is a correct distribution.
c) x | 3 | 6 | 8
P(x)| 0.2| 0 | 1
Solution: here the sum is 1.2 which is not equal to 1 .So isn't a probability distribution