# Statistical Methods-Bivariate Distributions (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 5 of 5

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## Question number: 1

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2015

### Describe in Detail

Let (X, Y) be distributed as bivariate normal BVN (3, 1, 16, 25, 3/5). Calculate P (4 < Y < 11.84|X=7)

### Explanation

We had known that the conditional distribution of Y given X has mean and variance .

Using conditional mean and variance

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## Question number: 2

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2014

### Describe in Detail

The marginal distributions of X and Y are given in the following table:

X | 1 | 2 | Total | |

Y | 3 4 | ? ? | ? ? | 1/4 3/4 |

Total | 1/2 | 1/2 | 1 |

If the co-variance between X and Y is zero, find the cell probabilities and see whether X and Y are independent.

### Explanation

Given that covariance between X and Y is zero

This gives that f (xy) =f (x) f (y)

So, the joint probability density function of x = 1, y = 3 is

We know that if the covariance is zero, then the conditional distribution is equal to unknown distribution

Similarly, the other will obtained.

x | 1 | 2 | Total | |

y | 3 | 1/8 | 1/… |

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## Question number: 3

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2013

### Describe in Detail

Let (X, Y) have a joint probability mass function

= 0, elsewhere

Find the marginal mass functions of X and Y.

### Explanation

Let (X, Y) have a joint probability mass function, the individual distribution of either X or Y is called the marginal distribution. So, the marginal mass functions of X is

and other values of f (x, y) is 0

The marginal mass functions of Y is

and other values of f (x, y) is 0

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## Question number: 4

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2009

### Describe in Detail

Given the joint density of (X _{1}, X _{2}),

Find the marginal densities of X _{1} and X _{2}. Also find E [X _{1}].

### Explanation

The joint density is written as

The marginal distribution of X _{1}

The marginal distribution of X _{2 } is

The first integral is

So, the marginal distribution of X _{2 } is

The mean of X _{1} is

when we solve this, we get

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## Question number: 5

» Statistical Methods » Bivariate Distributions » Bivariate Normal Distribution

Appeared in Year: 2010

### Describe in Detail

Show that for discrete distribution β _{2} > 1

### Explanation

We have to prove that

By definition of Kurtosis

Let x _{1}, x _{2}, …, x _{n} are n observations in a set have frequency f _{1}, f _{2, } …, f _{n} and the mean of observation is , then

Assume

which is always true because this a variance which is always positive

So,

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