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# Distributions

Understanding distributions is essential in statistics. They allow us to describe the behaviour in the results of different phenomena or experiments. And shed a light on what results we could expect and how probable they can be.

## Probability distributions

A probability distribution describes the entire set of outcomes that a random variable can take in a sample space. A distribution can either be discrete or continuous; however, the sum of all possible outcomes must sum to one.

A discrete random variable has a finite number of values that can be obtained and can be described by a probability mass function. This function will describe the likelihood of any of the values occurring.

For a fair six-sided die, the probability mass function is given as

 x

A discrete distribution can also be described by a cumulative probability function. This describes the probability of events up to that point happening, or .

For a fair six-sided die, the cumulative probability function is given as

 x 1 2 3 5 1

In the case of a continuous random variable, we can no longer describe things in this way, as we do not have a finite number of values. This means that the probability of a single event is zero, but we can find the probability that it obtains a value in a given range.

We describe a continuous distribution using a probability density function, normally named . To satisfy the criteria to be a probability distribution, we need the area under this function to be equal to one in the given sample space. To find the probability of an event happening , we evaluate the integral .

Suppose we have a sample space of [0,1], and we have a continuous random variable X, with a probability density function of . Find the probability of .

## The binomial distribution

When conducting an experiment with several trials, we can model the number of successful trials with a discrete random variable.

We can model the number of successful trials, X, with a binomial distribution, if certain conditions are met:

• A fixed number of trials (n).

• Two possible outcomes (success or failure).

• A fixed probability of success (p).

• All trials are independent.

If a random variable X has a binomial distribution, we can write , and this will have the probability mass function of

.

X is distributed binomially with n = 5 and p = 0.25. Find .

X is distributed binomially with n = 5 and p = 0.25. Find .

## The normal distribution

The normal distribution is continuous and is given by the probability density function , which has population parameters (representing the population mean) and (the population variance), with denoting the standard deviation. You do not yet possess the maths to show this, but we can show . Luckily, to find probabilities of intervals, you can use both your calculator and statistical tables.

A normal distribution is symmetrical and represented by a bell curve, asymptotic at both ends. Its mean is equal to its mode and median. About 68% of data is within one standard deviation from the mean, and this becomes 95% for two standard deviations and 99.7% for three standard deviations. If we plot the normal distribution, it looks like this.

The normal distribution 'bell curve'

If a random variable X is distributed binomially, with mean and variance , then we write .

Example:

X is a random variable, and . Find the standard deviation, and then find P (10 <x <15).

If the variance is 25 ( ), then the standard deviation ( ) is given as 5.

### The inverse normal

As suggested by the name, the inverse normal is used in the opposite way to the normal. You are given a region for the normal distribution and then use that to work out the probability. However, with the inverse normal, you are given the probability and expected to find the associated region. This is done using the inverse normal distribution on your calculator.

X is a continuous random variable, and . Find a if .

Putting this into a calculator, you will obtain a = 8.097602.

### The standard normal

We can standardize our normal distribution so that it has a mean of 0 and a standard deviation of 1. We can do this by coding the data as . This gives a set of z-values. The probability can be written as . Some values for can be found in statistical tables.

The random variable . Write in terms of For Example at unknown: .

### Approximating the binomial distribution

In a situation where we have , if n is sufficiently large, and p is sufficiently close to 0.5, then the binomial can be approximated by the normal distribution, where we denote and .

When calculating probabilities using this approximation, note that a continuity correction must be applied as we transition from a continuous distribution to a discrete one.

For example, if we have the random variable , which was being approximated by , we would apply the following continuity corrections:

Suppose . Use a normal approximation to approximate .

If we want Y to approximate X with a normal approximation, we have and

To find this probability, we need to use a continuity correction.

## Distribution - key takeaways

• A distribution describes the set of outcomes of a random variable.
• A probability mass function a discrete random variable, whereas a probability density function describes a continuous random variable.
• The binomial distribution is a discrete distribution with n trials and a chance of success.
• The conditions for the binomial distribution are: two possible outcomes, a fixed number of trials, a fixed outcome of success, and all trials are independent.
• A normal distribution is a continuous distribution with a population mean and population variance.
• The inverse normal distribution allows you to work out a region of the distribution, given a probability.
• The standard normal distribution has a mean of 0 and standard deviation of 1. We reach this from the normal by coding
• We can write as
• We can use the normal distribution to approximate a binomial distribution, so long as the chance of success is close to 0.5 and the number of trials is large.
• When approximating the binomial distribution using the normal and and when calculating probabilities, we must use a continuity correction.

The normal distribution is the set of possible outcomes of a continuous random variable when the population mean is μ and the population variance σ^2.

The binomial distribution is the set of possible outcomes of a discrete random variable with n trials and p chance of success.

A probability distribution describes all the possible outcomes of a random variable.

## Final Distributions Quiz

Question

What is probability distribution?

A probability distribution is the function that gives the individual probabilities of occurrence of different possible outcomes for an experiment.

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Question

What is the sum of all the probabilities of a probability distribution?

1

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The amount of rainfall in your city in March.

Continuous

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of trophies your favourite football club will win this season.

Discrete

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of students in the class who will pass the mathematics exam.

Discrete

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The weight of a newborn baby.

Continuous

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Question

Identify whether the following requires a discrete or continuous probability distribution?
The number of runs Joe Root will score in his next match

Discrete

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Question

The table for a probability distribution function has only 1 entry. What is the probability of that entry?

1

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Question

For a random variable X, what does the cumulative probability, P(X ≤ 8) represent?

The probability that the outcome of the random variable is less than or equal to 8.

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Question

A biased dice with faces numbered from 1 to 6 is rolled. The number obtained is modelled as a random variable X. Given that P (X = x) = k/x, find the value of k.

k/1 + k/2 + k/3 + k/4 +k/5 + k/6 = 1

=> k = 20/49

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Question

The discrete random variable X has the probability function,

P(X = x) = 0.2, x = 0,1= a, x = 2, 3=0.3, x = 4

Find the value of a.

0.2 + 0.2 + a + a + 0.3 = 1

=> a = 0.15

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Question

An unbiased coin is tossed 3 times and X is the random variable counting the number of heads obtained. What is P (X ≤ 2)?

0.875

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Question

The discrete random variable X has the probability function,

P(X = x) = kx, x = 1,2

= 0.2, x = 3,4

Find the value of k.

k + 2k + 0.2 × 2 = 1

=> k = 0.2

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Question

An experiment with 10 possible outcomes results in a uniform probability distribution. What is the probability of each outcome?

0.1

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Question

Which of the following is used for a discrete probability distribution?

probability mass function

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Question

Which of the following is used for a continuous probability distribution?

probability density function

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Question

There are 100 students in a Mathematics class. The random variable X represents the number of students taller than 6 feet. The cumulative probability at X=20 is 0.85. What is the probability that there are more than 20 students taller than 6 feet?

0.15

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Question

Can a binomial distribution be used for the following trial: the score when a fair dice is rolled.

No

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Question

Can a binomial distribution be used for the following trial: an unbiased coin is tossed.

Yes

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Question

Can a binomial distribution be used for the following trial: whether the score when a fair dice is rolled is greater than 4.

Yes

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Question

What are the necessary conditions for a binomial distribution?

1) there are a fixed number of trials, n

2)there are 2 possible outcomes, success and failure

3) there is a fixed probability of success, p, for all trials

4) the trials are independent

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Question

Which of the following can be used for a binomial distribution?

probability mass function

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Question

If a random variable X has the binomial distribution B(n, p), write down its probability mass function.

P(X = r) = nCr p^r (1 - p)^(n-r)

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Question

For the random variable X ~ B (5, 0.2), find P (X = 3).

0.0512

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Question

For the random variable X ~ B (5, 0.2), find P (X = 1).

0.4096

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Question

For the random variable X ~ B (5, 0.2), find P (X = 4).

0.0064

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Question

For the random variable X ~ B (9, 0.9), find P (X = 5).

0.0074

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Question

For the random variable X ~ B (9, 0.9), find P (X = 7).

0.172

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Question

For the random variable X ~ B (500, 0.4), what is the cumulative probability P (X ≤ 500)?

1

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Question

A fair dice is tossed five times. We build a binomial distribution to X ~ B(5, 0.5) to calculate the probability of getting x heads. What does the cumulative probability P (X ≤ 3) represent?

The probability of getting at most three heads

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Question

A fair dice is tossed 5 times. We build a binomial distribution to X ~ B(5, 0.5) to calculate the probability of getting x heads. What does (1 - P (X ≤ 3)) represent where P (X ≤ 3) is the cumulative probability at x=3?

the probability of getting more than three heads

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Question

State whether the following statement is true or false: the curve for a binomial distribution function is always symmetrical about its peak.

False

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Question

State whether the following statement is true or false: the sum of all probabilities of a binomial distribution can be negative.

False

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Question

What is the normal distribution?

The normal distribution is a continuous probability distribution that can be presented on a graph.

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Question

What does the normal distribution represent? Please give examples.

The normal distribution represents continuous random variables, such as, height, weight and measurement errors.

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Question

What is the normal distribution curve?

The normal distribution curve is a bell shaped curve that is symmetrical to the mean.

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Question

What can the standard deviation tell you?

The standard deviation tells you how spread out the data is.

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Question

What does it mean when the data falls within one standard deviation of the mean?

68% of the data falls here, the probability is likely.

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Question

What does it mean when the data falls within two standard deviations of the mean?

95% of the data falls here, the probability is very likely.

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Question

What does it mean when the data falls within three standard deviations of the mean?

99.7% of the data falls here, the probability is almost certain.

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Question

What is the mean of a standard normal distribution?

0 (zero)

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Question

What is the standard deviation of a standard normal distribution?

1

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Question

The IQs of students at a school are normally distributed with standard deviation 15. If 20% of students have an IQ above 125, find the mean IQ of students at that school.

112

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Question

The arrival times of buses are normally distributed with standard deviation 5min. If 10% of the buses arrive before 3:55pm, find the mean arrival time of the buses.

4:01:24pm

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Question

The results of a test are normally distributed. Harry's z-score is -2. What does this signify?

Harry's score is 2 standard deviations below the class mean.

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