Example: Tossing a coin: we can gain Heads or Tails.

Let"s provide them the worths Heads=0 and also Tails=1 and we have a Random Variable "X":

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So:

We have actually an experiment (prefer tossing a coin)We give values to each eventThe set of values is a Random Variable

Discover even more at Random Variables.

You are watching: Find the mean, variance and standard deviation for the probability distribution given below:

Median, Variance and also Standard Deviation


Example: Tossing a single unfair die

For fun, imagine a weighted die (cheating!) so we have actually these probabilities:

123456
0.10.10.10.10.10.5

Median or Expected Value: μ

When we recognize the probability p of eexceptionally value x we have the right to calculate the Expected Value (Mean) of X:


Note: Σ is Sigma Notation, and also indicates to amount up.

To calculate the Expected Value:

multiply each worth by its probabilitysum them up

Example continued:

x123456
p0.10.10.10.10.10.5
xp0.10.20.30.40.53

μ = Σxp = 0.1+0.2+0.3+0.4+0.5+3 = 4.5

The meant worth is 4.5


Note: this is a weighted mean: worths with better probcapability have higher contribution to the intend.

Variance: Var(X)

The Variance is:


To calculate the Variance:

square each value and multiply by its probabilitysum them up and we gain Σx2pthen subtract the square of the Expected Value μ2

Example continued:

x123456
p0.10.10.10.10.10.5
x2p0.10.40.91.62.518

Σx2p = 0.1+0.4+0.9+1.6+2.5+18 = 23.5

Var(X) = Σx2p − μ2 = 23.5- 4.52 = 3.25

The variance is 3.25


Example continued:

x123456
p0.10.10.10.10.10.5
x2p0.10.40.91.62.518

σ = √Var(X) = √3.25 = 1.803...

The Standard Deviationis 1.803...


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You arrangement to open up a brand-new McDougals Fried Chicken, and also uncovered these stats for comparable restaurants:

PercentYear"s Earnings
20%$50,000 Loss
30%$0
40%$50,000 Profit
10%$150,000 Profit

Using that as probabilities for your new restaurant"s profit, what is the Expected Value and Standard Deviation?

The Random Variable is X = "possible profit".

Sum up xp and x2p:


ProbabilitypWages ($"000s)xxpx2p
0.2-50-10500
0.3000
0.450201000
0.1150152250
Σp = 1Σxp = 25 Σx2p = 3750

μ = Σxp = 25

Var(X) = Σx2p − μ2 = 3750 − 252 = 3750 − 625 = 3125

σ = √3125 = 56 (to nearemainder entirety number)

But remember these are in countless dollars, so:

μ = $25,000σ = $56,000

So you might suppose to make $25,000, yet with a very wide deviation feasible.


Example (continued):

Now via different probabilities (the $50,000 value has a high probcapacity of 0.7 now):


ProbabilitypIncomes ($"000s)xxpx2p
0.1-50-5250
0.1000
0.750351750
0.1150152250
Σp = 1Sums:Σxp = 45Σx2p = 4250

μ = Σxp = 45

Var(X) = Σx2p − μ2 = 4250 − 452 = 4250 − 2025 = 2225

σ = √2225 = 47 (to nearemainder whole number)

In thousands of dollars:

μ = $45,000σ = $47,000

The intend is now a lot closer to the many probable worth.

See more: Which Part Of The Brain Is The &Quot;Executive Suite&Quot; For All Brain Activity?

And the standard deviation is a little smaller (showing that the worths are more central.)


Continuous

Random Variables have the right to be either Discrete or Continuous:

Discrete Data deserve to just take particular worths (such as 1,2,3,4,5) Continuous File deserve to take any type of worth within a selection (such as a person"s height)

Here we looked only at discrete data, as finding the Typical, Variance and also Standard Deviation of constant data demands Integration.

Summary


A Random Variable is a variable whose feasible values are numerical outcomes of a random experiment.The Mean (Expected Value) is: μ = ΣxpThe Variance is: Var(X) = Σx2p − μ2The Standard Deviation is: σ = √Var(X)