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Probability Distribution

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Probability Distribution

A probability distribution is a function that gives the individual probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events.

Expressing a probability distribution

A probability distribution is often described in the form of an equation or a table that links each outcome of a probability experiment with its corresponding probability of occurring.

Example 1

Consider an experiment where the random variable X = the score when a fair dice is rolled.

Since there are six equally likely outcomes here, the probability of each outcome is 1/6.

The corresponding probability distribution can be described:

  • As a probability mass function:

P (X = x) = 1/6, x = 1, 2, 3, 4, 5, 6

  • In the form of a table:

x

1

2

3

5

P (X = x)

1/6

1/6

1/6

1/6

1/6

1/6

Example 2

A fair coin is tossed twice in a row. X is defined as the number of heads obtained. Write down all the possible outcomes, and express the probability distribution as a table and as a probability mass function.

Solution 2

With heads as H and tails as T, there are 4 possible outcomes:

(T, T), (H, T), (T, H) and (H, H).

Therefore the probability of getting (X = x = number of heads = 0) = (number of outcomes with 0 heads) / (total number of outcomes) = 1/4

(x = 1) = (number of outcomes with 1 heads) / (total number of outcomes) = 2/4

(x = 2) = (number of outcomes with 2 heads) / (total number of outcomes) = 1/4

Now let's express the probability distribution

  • As a probability mass function:

P (X = x) = 0.25, x = 0, 2

= 0.5, x = 1

  • In the form of a table:

No. of heads, x

0

1

2

P (X = x)

0.25

0.5

0.25

Example 3

The random variable X has a probability distribution function

P (X = x) = kx, x = 1, 2, 3, 4, 5

What is the value of k?

Solution 3

We know that the sum of the probabilities of the probability distribution function has to be 1.

For x = 1, kx = k.

For x = 2, kx = 2k.

And so on.

Thus, we have

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

=> k = 1/15

Discrete and continuous probability distribution

Probability distribution functions can be classified as discrete or continuous depending on whether the domain takes a discrete or a continuous set of values.

Discrete probability distribution function

Mathematically, a discrete probability distribution function can be defined as a function p (x) that satisfies the following properties:

1. The probability that x can take a specific value is p (x). That is

P (X = x) = p (x) = px

2. p (x) is non-negative for all real x.

3. The sum of p (x) over all possible values of x is 1, that is

= 1

A discrete probability distribution function can take a discrete set of values – they need not necessarily be finite. The examples we have looked at so far are all discrete probability functions. This is because the instances of the function are all discrete – for example, the number of heads obtained in a number of coin tosses. This will always be 0 or 1 or 2 or… You will never have (say) 1.25685246 heads and that is not part of the domain of that function. Since the function is meant to cover all possible outcomes of the random variable, the sum of the probabilities must always be 1.

Further examples of discrete probability distributions are:

  • X = the number of goals scored by a football team in a given match.

  • X = the number of students who passed the mathematics exam.

  • X = the number of people born in the UK in a single day.

Discrete probability distribution functions are referred to as probability mass functions.

Continuous probability distribution function

Mathematically, a continuous probability distribution function can be defined as a function f (x) that satisfies the following properties:

1. The probability that x is between two points a and b is

p (a ≤ x ≤ b) = f (x) dx

2. It is non-negative for all real x.

3. The integral of the probability function is one that is

f (x) dx = 1

A continuous probability distribution function can take an infinite set of values over a continuous interval. Probabilities are also measured over intervals, and not at a given point. Thus, the area under the curve between two distinct points defines the probability for that interval. The property that the integral must be equal to one is equivalent to the property for discrete distributions that the sum of all the probabilities must be equal to one.

Examples of continuous probability distributions are:

X = the amount of rainfall in inches in London for the month of March.

X = the lifespan of a given human being.

X = the height of a random adult human being.

Continuous probability distribution functions are referred to as probability density functions.

Cumulative probability distribution

A cumulative probability distribution function for a random variable X gives you the sum of all the individual probabilities up to and including the point x for the calculation for P (X ≤ x).

This implies that the cumulative probability function helps us to find the probability that the outcome of a random variable lies within and up to a specified range.

Example 1

Let's consider the experiment where the random variable X = the number of heads obtained when a fair dice is rolled twice.

The cumulative probability distribution would be the following:

No. of heads, x

0

1

2

P (X = x)

0.25

0.5

0.25

Cumulative Probability

P (X ≤ x)

0.25

0.75

1

The cumulative probability distribution gives us the probability that the number of heads obtained is less than or equal to x. So if we want to answer the question, “what is the probability that I will not get more than heads”, the cumulative probability function tells us that the answer to that is 0.75.

Example 2

A fair coin is tossed three times in a row. A random variable X is defined as the number of heads obtained. Represent the cumulative probability distribution using a table.

Solution 2

Representing obtaining heads as H and tails as T, there are 8 possible outcomes:

(T, T, T), (H, T, T), (T, H, T), (T, T, H), (H, H, T), (H, T, H), (T , H, H) and (H, H, H).

The cumulative probability distribution is expressed in the following table.

No. of heads, x

0

1

2

3

P (X = x)

0.125

0.375

0.375

0.125

Cumulative Probability

P (X ≤ x)

0.125

0.5

0.875

1

Example 3

Using the cumulative probability distribution table obtained above, answer the following question.

  1. What is the probability of getting no more than 1 head?

  2. What is the probability of getting at least 1 head?

Solution 3

  1. The cumulative probability P (X ≤ x) represents the probability of getting at most x heads.

Therefore, the probability of getting no more than 1 head is P (X ≤ 1) = 0.5

  1. The probability of getting at least 1 head is 1 - P (X ≤ 0) = 1 - 0.125 = 0.875

Uniform probability distribution

A probability distribution where all of the possible outcomes occur with equal probability is known as a uniform probability distribution.

Thus, in a uniform distribution, if you know the number of possible outcomes is n probability, the probability of each outcome occurring is 1 / n.

Example 1

Let us get back to the experiment where the random variable X = the score when a fair dice is rolled.

We know that the probability of each possible outcome is the same in this scenario, and the number of possible outcomes is 6.

Thus, the probability of each outcome is 1/6.

The probability mass function will therefore be,

P (X = x) = 1/6, x = 1, 2, 3, 4, 5, 6

Binomial probability distribution

Binomial Distribution is a probability distribution function that is used when there are exactly two mutually exclusive possible outcomes of a trial. The outcomes are classified as "success" and "failure", and the binomial distribution is used to obtain the probability of observing x successes in n trials.

Intuitively, it follows that in the case of a binomial distribution, the random variable X can be defined to be the number of successes obtained in the trials.

You can model X with a binomial distribution, B (n, p), if:

  • there are a fixed number of trials, n

  • there are 2 possible outcomes, success and failure

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

  • the trials are independent

Probability Distribution - Key takeaways

    • A probability distribution is a function that gives the individual probabilities of occurrence of different possible outcomes for an experiment. Probability distributions can be expressed as functions as well as tables.

    • Probability distribution functions can be classified as discrete or continuous depending on whether the domain takes a discrete or a continuous set of values. Discrete probability distribution functions are referred to as probability mass functions. Continuous probability distribution functions are referred to as probability density functions.

    • A cumulative probability distribution function for a random variable X gives you the sum of all the individual probabilities up to and including the point, x, for the calculation for P (X ≤ x).

    • A probability distribution where all of the possible outcomes occur with equal probability is known as a uniform probability distribution. In a uniform probability distribution, if you know the number of possible outcomes, n, the probability of each outcome occurring is 1 / n.

Frequently Asked Questions about Probability Distribution

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

To find the mean of a probability distribution, we multiply the value of each outcome of the random variable with its associated probability, and then find the mean of the resultant values.

A discrete probability distribution fulfils the following requirements : 1) The probability that x can take a specific value is p(x). That is P[X = x] = p(x) = px 2) p(x) is non-negative for all real x. 3) The sum of p(x) over all possible values of x is 1.

A binomial distribution is a probability distribution that is used when there are exactly two mutually exclusive possible outcomes of a trial. The outcomes are classified as "success" and "failure", and the binomial distribution is used to obtain the probability of observing x successes in n trials.

In an uniform distribution probability function, each outcome has the same probability. Thus, if you know the number of possible outcomes, n, the probability for each outcome is 1/n.

Final Probability Distribution Quiz

Question

What is probability distribution?

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Answer

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

Show question

Question

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

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Answer

1

Show question

Question

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

Show answer

Answer

Continuous

 

Show question

Question

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

Show answer

Answer

Discrete

Show question

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.

Show answer

Answer

Discrete

Show question

Question

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

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Answer

Continuous

Show question

Question

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

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Answer

Discrete

Show question

Question

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

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Answer

1

Show question

Question

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

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Answer

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.

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Answer

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

=> k = 20/49

Show question

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.

Show answer

Answer

0.2 + 0.2 + a + a + 0.3 = 1

=> a = 0.15

Show question

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)?

Show answer

Answer

0.875

Show question

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.

Show answer

Answer

k + 2k + 0.2 × 2 = 1

=> k = 0.2

Show question

Question

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

Show answer

Answer

0.1

Show question

Question

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

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Answer

probability mass function

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Question

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

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Answer

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?

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Answer

0.15

Show question

Question

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

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Answer

No

Show question

Question

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

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Answer

Yes

Show question

Question

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

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Answer

Yes

Show question

Question

 What are the necessary conditions for a binomial distribution?

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Answer

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

Show question

Question

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

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Answer

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.

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Answer

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).

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Answer

0.0512

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Question

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

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Answer

0.4096

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Question

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

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Answer

0.0064

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Question

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

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Answer

0.0074

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Question

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

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Answer

0.172

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Question

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

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Answer

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?

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Answer

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?

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Answer

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.

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Answer

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.

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Answer

False

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Question

What is the normal distribution?

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Answer

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.

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Answer

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?


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Answer

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?


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Answer

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?


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Answer

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?


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Answer

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?


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Answer

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?

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Answer

0 (zero)

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Question

What is the standard deviation of a standard normal distribution?

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Answer

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.

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Answer

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.

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Answer

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?

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Answer

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

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