12 Linear Combinations

Question 1

What is the sum of all probabilities in any probability distribution?

Answer 1

1

This is a fundamental property of probability distributions.

Question 2

For DISCRETE RANDOM VARIABLES, the mean is given by __________.

Answer 2

E(X) = Σxp

This formula calculates the expected value of a discrete random variable.

Question 3

For DISCRETE RANDOM VARIABLES, the variance is given by __________.

Answer 3

Question 4

Two common examples of DISCRETE DISTRIBUTIONS are __________.

Answer 4

• Binomial distribution
• Poisson distribution

These distributions are widely used in statistics for different types of data.

Question 5

The continuous uniform distribution is sometimes referred to as the __________.

Answer 5

RECTANGULAR DISTRIBUTION

This distribution has constant probability across its range.

Question 6

A discrete random variable is such that each of its outcomes is assumed to be equally likely. This distribution is known as the __________.

Answer 6

Discrete Uniform distribution

Question 7

The linear combination of normal variables is also normally distributed if the variables are __________.

Answer 7

INDEPENDENT

Question 8

In the context of random variables, what does E(X) represent?

Answer 8

The expected value of the random variable X

Question 9

True or false: The mean of a Poisson distribution is equal to its variance.

Answer 9

TRUE

In a Poisson distribution, both parameters are equal.

Question 10

The mean of a normally distributed variable is represented by the symbol __________.

Answer 10

μ

This symbol denotes the average of the distribution.

Question 11

The variance of a normally distributed variable is represented by the symbol __________.

Answer 11

σ²

This symbol denotes the spread of the distribution.

Question 12

In a linear combination of independent normal random variables, the mean is calculated by __________.

Answer 12

Applying the combination to the means of the independent variables

Do everything to the mean

Question 13

What is the rule for finding the mean of a linear combination of independent variables?

Answer 13

Do everything

Question 14

When do you square the multiplier for the variance?

Answer 14

When the multiplier is not 1

That is when the varibales are the same

Question 15

What rules need to be applied to the variance for a combination of variables?

Answer 15

• Always ADD
• Affected by MULTIPLIERS ONLY

Question 16

𝑋~𝑁(15,3) and 𝑌~𝑁(6,2).
If 𝑋 and 𝑌 are independent, find the distribution of 𝐴 where 𝐴=𝑋+𝑌

Answer 16

A~N(21,5)

Question 17

𝑋~𝑁(15,3) and 𝑌~𝑁(6,2).
If 𝑋 and 𝑌 are independent, find the distribution of 𝐴 where 𝐴=4𝑋-2𝑌

Answer 17

A~N(48,56)

Question 18

Bottles of mineral water are delivered to shops in crates containing 12 bottles each. The weights of bottles are normally distributed with mean weight 2𝑘𝑔 and standard deviation 0.05𝑘𝑔. The weights of empty crates are normally distributed with mean 2.5𝑘𝑔 and standard deviation 0.3𝑘𝑔.
Assuming that all random variables are independent, find the probability that a full crate will weigh between 26𝑘𝑔 and 27𝑘𝑔.

Answer 18

T~N(26.5,0.12)
0.851

Question 19

Bottles of mineral water are delivered to shops in crates containing 12 bottles each. The weights of bottles are normally distributed with mean weight 2𝑘𝑔 and standard deviation 0.05𝑘𝑔. Two bottles are selected at random from a crate. Find the probability that they differ in weight by more than 0.1𝑘𝑔.

Answer 19

B1-B2~N(0,0.005)
0.157

Question 20

A machine creates two types of balls, the machine produces rubber balls whose diameters are normally distributed with mean 5.5cm and standard deviation 0.08cm and another rubber balls whose diameters are normally distributed with mean 3.2cm and standard deviation 0.02cm
What is the mean and standard deviation of the width of two such balls chosen and placed next to each other?

Answer 20

A+B~N(8.7,0.0068)

Question 21

A machine produces rubber balls whose diameters are normally distributed with mean 5.5cm and standard deviation 0.08cm.
The balls are packed into tubes with an internal diameter normally distributed with mean 5.7cm and standard deviation 0.12cm.
What is the probability that a rubber ball selected at random will not fit into the tube?

Answer 21

A>T, A-T>0
A-T~N(-0.2, 0.0208)
0.0828