Statistics involves collecting, analyzing, interpreting, and presenting data to draw meaningful conclusions. Probability is the study of the likelihood and patterns of events occurring, serving as the foundation for statistical inference.
Random Variable
Histogram
- is a function that associates a real number to each element in
the sample space
- After determining the probability distribution this is the next
step (coins)
- sample space should always be equal to 1 (1 flat walang point
point) and then you will write it as p(x)=1
- if it’s not equal to 1 you will write p(x)≠1 (no need to solve
for mean, variants and standard deviation)
Discrete Random Variable
- whole number
Mean
Continuous Random Variable
- Denoted as µ
- With decimal - Formulas: 𝜇 = 𝑥1 ∙ 𝑝(𝑥)1 + 𝑥2 ∙ 𝑝(𝑥)2 … . . 𝑥𝑛 ∙
𝑝(𝑥)𝑛
Probability Distribution
- 𝜇 = ∑ 𝑋 ∙ 𝑃(𝑥)
Example: Suppose three coins are tossed. Let Y be the random variable
representing the number of tails that occur. Find the probability of each of the Step 1: make a probability distribution table
values of the random variable Y.
Step 2: multiply the X to the p(x)
Step 1: determine the sample space
- 𝑥 ∙ 𝑝(𝑥)
S = {TTT, TTH, THT, HTT, HHT, HTH, THH, HHH}
Step 3: add all the answers obtained in step 2
Step 2: count the sample space that the instructions look for
Variants (variance)
Possible Variable y
Outcomes (number of tails - Denoted as 𝜎 2
TTT 3 - Formula: 𝜎 2 = ∑(𝑥 − 𝜇)2 𝑝(𝑥)
TTH 2
THT 2 Step 1: find the mean on the probability distribution table
HTT 2
Step 2: subtract the mean to the variable X (𝑥 − 𝜇)
HHT 1
HTH 1 Step 3: Square the results obtained in step 2 ( (𝑥 − 𝜇)2)
THH 1
HHH 0 Step 4: multiply the answers on step 3 to the P(x) ( (𝑥 − 𝜇)2 ∙ 𝑃(𝑥))
Step 3: make it a fraction Step 5: total the answers obtained in step 4
Number of 0 1 2 3 Standard Deviation
Tails (y)
Probability 1/8 3/8 3/8 1/8 - Denoted as 𝜎
P(y) - Formula: 𝜎 = √𝜎 2
- 𝜎 = √(𝑥 − 𝜇)2 ∙ 𝑃(𝑥)
3 coins = 8 outcomes
4 coins = 16 outcomes
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