PSLP

4rd SEMESTER

BS-202

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UNIT-2

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Moment Generating Functions

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Continuous Distributions

The Uniform distribution from a to b

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The Normal distribution �(mean μ, standard deviation σ)

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The Exponential distribution

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Weibull distribution with parameters α and β.

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The Weibull density, f(x)

(α = 0.5, β = 2)

(α = 0.7, β = 2)

(α = 0.9, β = 2)

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The Gamma distribution

Let the continuous random variable X have density function:

Then X is said to have a Gamma distribution with parameters α and λ.

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Expectation of functions of Random Variables

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X is discrete

X is continuous

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Moments of Random Variables

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The k^th moment of X.

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the k^th central moment of X

where μ = μ₁ = E(X) = the first moment of X .

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Some Rules:

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Moment generating functions

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Moment Generating function of a R.V. X

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Examples

1. The Binomial distribution (parameters p, n)

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The moment generating function of X , m_X(t) is:

2. The Poisson distribution (parameter λ)

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The moment generating function of X , m_X(t) is:

3. The Exponential distribution (parameter λ)

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The moment generating function of X , m_X(t) is:

4. The Standard Normal distribution (μ = 0, σ = 1)

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We will now use the fact that

We have completed the square

This is 1

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The moment generating function of X , m_X(t) is:

4. The Gamma distribution (parameters α, λ)

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We use the fact

Equal to 1

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Properties of� Moment Generating Functions

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1. m_X(0) = 1

Note: the moment generating functions of the following distributions satisfy the property m_X(0) = 1

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We use the expansion of the exponential function:

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Now

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Property 3 is very useful in determining the moments of a random variable X.

Examples

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To find the moments we set t = 0.

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The moments for the exponential distribution can be calculated in an alternative way. This is note by expanding m_X(t) in powers of t and equating the coefficients of t^k to the coefficients in:

Equating the coefficients of t^k we get:

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The moments for the standard normal distribution

We use the expansion of e^u.

We now equate the coefficients t^k in:

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If k is odd: μ_k = 0.

For even 2k:

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Summary

Moments

Moment generating functions

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Moments of Random Variables

The moment generating function

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Examples

1. The Binomial distribution (parameters p, n)

2. The Poisson distribution (parameter λ)

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3. The Exponential distribution (parameter λ)

4. The Standard Normal distribution (μ = 0, σ = 1)

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5. The Gamma distribution (parameters α, λ)

6. The Chi-square distribution (degrees of freedom ν)

(α = ν/2, λ = 1/2)

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1. m_X(0) = 1

Properties of Moment Generating Functions

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Let l_X (t) = ln m_X(t) = the log of the moment generating function

The log of Moment Generating Functions

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Thus l_X (t) = ln m_X(t) is very useful for calculating the mean and variance of a random variable

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Examples

1. The Binomial distribution (parameters p, n)

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2. The Poisson distribution (parameter λ)

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3. The Exponential distribution (parameter λ)

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4. The Standard Normal distribution (μ = 0, σ = 1)

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5. The Gamma distribution (parameters α, λ)

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6. The Chi-square distribution (degrees of freedom ν)

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Box and Whisker Plots

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Order numbers

3, 5, 4, 2, 1, 6, 8, 11, 14, 13, 6, 9, 10, 7

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• First, order your numbers from least to greatest:

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1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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Median

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Then find the median (from the ordered list):

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• Cross off one number from each side until you reach the middle number (or numbers).

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1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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Median (continued):

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• If there are two numbers in the middle,

Add those 2 middle numbers together:

6 + 7 = 13

• Then divide by 2:

13 ÷ 2 = 6.5

• The median is 6.5.

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Quartiles (page 1)

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Then split the numbers on left and right sides of the median:

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1, 2, 3, 4, 5, 6, 6, │7, 8, 9, 10, 11, 13, 14

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Quartiles (page 2)

1, 2, 3, 4, 5, 6, 6, │7, 8, 9, 10, 11, 13, 14

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• Find the median for each half:

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1, 2, 3, 4, 5, 6, 6 │ 7, 8, 9, 10, 11, 13, 14

1, 2, 3, 4, 5, 6, 6 │ 7, 8, 9, 10, 11, 13, 14

Left Right

Median = 4 Median = 10

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Quartiles (page 3)

1, 2, 3, 4, 5, 6, 6 │ 7, 8, 9, 10, 11, 13, 14

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Left Right

Median = 4 Median = 10

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• The left median is called the LOWER QUARTILE.
• The right median is called the UPPER QUARTILE.

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Number line

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Draw a number line from the smallest to the largest number without skipping any numbers.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Quartiles on number line

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Put circles at the LOWER and UPPER Quartiles.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Box on Quartiles on number line

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Draw a box connecting the circles at the LOWER and UPPER Quartiles.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Median on number line

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Put a circle at the median (6.5).

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Median on number line

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Draw a line connecting the median to the box.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Low and high numbers

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Put circles at the high and low points.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Low and high numbers

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11, 13, 14

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• Draw lines that connect the high and low points to the box.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Box and Whisker Plot

3, 5, 4, 2, 1, 6, 8, 11, 14, 13, 6, 9, 10, 7

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1 2 3 4 5 6 7 8 9 10 11 12 13 14

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Here is the completed Box and Whisker Plot!

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Ques. The number of books taken out of the library per month by first year students from a sample of 15 is as follows: 3, 0, 12, 0, 2, 0, 26, 0, 7, 5, 5, 2, 1, 1, 2.

The number of books taken out of the library per month by third year students from a sample of 15 is as follows: 12, 0, 9, 4, 15, 2, 6, 10, 27, 15, 5, 9, 1, 14, 2.

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Question?

The number of books taken out of the library per month by first year students from a sample of 15 is as follows: 3, 0, 12, 0, 2, 0, 26, 0, 7, 5, 5, 2, 1, 1, 2.

The number of books taken out of the library per month by third year students from a sample of 15 is as follows: 12, 0, 9, 4, 15, 2, 6, 10, 27, 15, 5, 9, 1, 14, 2.

Solution:

First of all start by ordering the data.�The number of books taken out of the library per month by first year students from a sample of 15 is as follows:0, 0, 0, 0, 1, 1, 2, 2, 2, 3, 5, 5, 7, 12, 2.

The number of books taken out of the library per month by third year students from a sample of 15 is as follows:0, 1, 2, 2, 4, 5, 6, 9, 9, 10, 12, 14, 15, 15, 27.

For first year students the data is as follows:�Sample size: 15�Median: 2�Minimum value: 0�Maximum value: 26�First quartile: 0�Third quartile: 5�Interquartile Range: 5�Outliers: 26�For third year students the data is as follows:�Sample size: 15�Median: 9�Minimum value 0�Maximum value: 27�First quartile: 2�Third quartile: 14�Interquartile Range: 12�Outliers: none.

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Solution:

For first year students the data is as follows:�Sample size: 15�Median: 2�Minimum value: 0�Maximum value: 26�First quartile: 0�Third quartile: 5�Interquartile Range: 5�Outliers: 26�For third year students the data is as follows:�Sample size: 15�Median: 9�Minimum value 0�Maximum value: 27�First quartile: 2�Third quartile: 14�Interquartile Range: 12�Outliers: none.

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Covariance and Correlation

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As an example, take g(x, y) = xy for discrete random variables X and Y with

the joint probability distribution given in the table. The expectation of XY is computed as follows:

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Proof that E[X + Y] = E[X] + E[Y]:

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Var(X + Y) is generally not equal to Var(X) + Var(Y)

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If Cov(X,Y) > 0 , then X and Y are positively correlated.

If Cov(X,Y) < 0, then X and Y are negatively correlated.

If Cov(X,Y) =0, then X and Y are uncorrelated.

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Now let X and Y be two independent random variables.

Then Cov(X, Y ) = E[XY ] − E[X]E[Y ] = 0.

Hence, then X and Y are uncorrelated.

We proved that if X and Y are two independent random variables,

then they are uncorrelated.

In general, E[XY] is NOT equal to E[X]E[Y].

INDEPENDENT VERSUS UNCORRELATED.

If two random variables X and Y are independent, then X and Y are uncorrelated.

The converse is not true as we will see on the next slide.

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Here, Cov =0 but P (X, Y) not equals P(X). P(Y). Events are not independent.

Then Cov(X, Y ) = E[XY ] − E[X]E[Y ] = 0 and X and Y are uncorrelated,�but they are dependent.

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The variance of a random variable with a Bin(n,p) distribution:

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The covariance changes under a change of units

The covariance Cov(X,Y) may not always be suitable to express the dependence between X and Y. For this reason, there is a standardized version of the covariance called the correlation coefficient of X and Y, which remains unaffected by a change of units and, therefore, is dimensionless.

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Correlation coefficient is also called Pearson correlation coefficient.

(from Wikipedia) Examples of scatter diagrams with different values of correlation coefficient.

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(from Wikipedia) Several sets of (x, y) points, with the correlation coefficient of x and y for each set. Note that the correlation reflects the non-linearity and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of Y is zero.

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SUMMARY:

• Covariance and correlation are both measures used in statistics to describe the relationship between two variables. However, they serve slightly different purposes and have different properties.

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Covariance:�

• Covariance measures the degree to which two variables change together.
• It indicates the direction of the linear relationship between two variables.
• The sign of covariance (positive or negative) indicates the direction of the relationship: positive covariance means the variables tend to move in the same direction, while negative covariance means they tend to move in opposite directions.
• The magnitude of covariance is not standardized, so it's difficult to interpret the strength of the relationship. It can be affected by the scale of the variables.

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• The formula for covariance between two variables X and Y is:
• Cov(X, Y)=∑(Xi-a)(Y-b)/(N-1)

Where Xi​ and Yi​ are individual data points, a and b are the means of X and Y respectively, and

N is the number of data points.

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Correlation

• Correlation is a standardized measure of the relationship between two variables.
• It not only indicates the direction of the relationship but also quantifies the strength of the relationship between variables.
• Correlation always lies between -1 and 1.
• A correlation coefficient of 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship.
• Correlation is unaffected by changes in scale, making it easier to compare the strength of relationships between different pairs of variables.

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• The formula for the Pearson correlation coefficient (the most commonly used correlation coefficient) between two variables X and Y is:

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cor(X,Y)=cov(X,Y)/σX​⋅σY​

Where, cov(X,Y) is the covariance between X and Y, and σX​ and σY​ are the standard deviations of X and Y respectively.

σX= square root [(Xi-a)^2/n], a=mean of Xi

NOTE: While covariance and correlation both describe the relationship between variables, correlation is a more useful measure because it's standardized and provides a clear indication of the strength and direction of the relationship.

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��CENTRAL LIMIT THEOREM

• Central Limit Theorem plays a very central role in the study of statistics. It has relevance in hypothesis testing and confidence interval.
• It characterizes the convergence towards normal distribution. As the sample size increases, the distribution of the sample mean (or sum) approaches a limiting distribution, called as normal distribution. This limiting behavior is the term "limit" refers to in the theorem's name.

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The Central Limit Theorem describes the properties of the following two quantities as n gets larger

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The sample mean or average of a random sample of size n:

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The sum or total of the sample of size n:

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Let’s take an example:

• Population of 65 children showing their marks

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30 32 32 33 33 34 38 43

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Samples

• Taking a random sample of size n=1 from the population.
• Sample mean of sample of size n=1 is number itself.

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Sample of Size 1

• Take a random sample of size n=1 from the population.
• Sample mean of sample of size n=1 is

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Another Sample of Size 1

• Take a random sample of size n=1 from the population.
• Sample mean of sample of size n=1 is

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Central Limit Theorem

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No matter the shape of the population, the distribution of x-bars tends toward Normality

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Example

Suppose that the mean time for an oil change at an oil change joint is 11.4 minutes with a standard deviation of 3.2 minutes.

1. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean.

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(b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes?

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Example

Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes.

1. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean.

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Solution: is approximately normally distributed

with mean = 11.4 and std. dev. = .

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Example

Suppose that the mean time for an oil change at a “10-minute oil change joint” is 11.4 minutes with a standard deviation of 3.2 minutes.

1. If a random sample of n = 35 oil changes is selected, describe the sampling distribution of the sample mean.

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(b) If a random sample of n = 35 oil changes is selected, what is the probability the mean oil change time is less than 11 minutes?

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Solution: is approximately normally distributed

with mean = 11.4 and std. dev. = .

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Solution: , P(Z < –0.74) = 0.23.

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Another Example:

• Suppose we have a fair six-sided die. When rolled, each side has an equal probability of 1/6 of showing up. Let's denote the result of rolling this die as a random variable X.
• Now, let's say we roll this die n times independently and record the sum of the outcomes. We can denote each outcome of rolling the die as X1​, X2​,…,Xn​, where each Xi​ represents the result of the ith roll.

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