Question

Let u and v be column vectors containing real numbers. Describe how to calculate the dot product of u and v.

What is the difference between a discrete random variable and a continuous random variable?

data science

Answer 1

1)

The dot product of vectors u = {u_x ; u_y ; u_z} and v = {v_x ; v_y ; v_z} can be figured out by using the following formula for the spatial problem:

u · v = u_x · v_x + u_y · v_y + u_z · u_z

Dot product formula for n dimensional space problems

Talking about the n dimensional space problem, the dot product of vectors u = {u₁ ; u₂ ; … ; u_n} and v = {v₁ ; v₂ ; … ; v_n} can be easily determined with the following formula:

u · v = u₁ · v₁ + u₂ · v₂ + … + u_n · v_n

2)

The difference b/w discrete random variable and continuous random variable is very similar to digital and analog signals

In digital signals -the signal can only attain ‘discrete’ values

discrete means individual or distinct like 0 and 1.

for e.g. DC waveform

A real life example would be to count no of students sitting in a class,no of marbles in a jar

analog signal-an analog signal is the one whose value keeps changing at every instant of time

for e.g. Sine WAVE

A real life example would be finding the height on persons ,or finding the total no of possible values that can be displayed by a mechanical clock while moving from 12 to 1 position or 1 to 2 and so on

Coming to mathematical Point Of View

A discrete random variable X has a countable number of possible values.

Example: Let X represent the sum of two dice.

Then the probability distribution of X is as follows:

X P(X)

2 1/36

3 2 /36

4 3/36

5 4/36

6 5/36

7 6/36

8 5/36

9 4/36

10 3/36

11 2/36

12 1/36

To graph the probability distribution of a discrete random variable, construct a probability histogram.

A continuous random variable X takes all values in a given interval of numbers.

▪ The probability distribution of a continuous random variable is shown by a density curve.

▪ The probability that X is between an interval of numbers is the area under the density curve between the interval endpoints

▪ The probability that a continuous random variable X is exactly equal to a number is zero