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Random Variables and Probability Distributions — Flashcards

Telangana Open School (TOSS) · Class 12 · Mathematics

25 flashcards for Random Variables and Probability Distributions (Telangana Open School (TOSS) Class 12 Mathematics) to test yourself on key terms.

49 questions25 flashcards9 formulas & key relations5 concepts

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25 Flashcards·
Discrete Random VariablesMean of Random VariableVariance of Random VariableProbability DistributionBinomial Distribution
Card 1Discrete Random Variables

A fair die is rolled once. Let X be the number on the face. Find P(X = 4).

Answer

Since the die is fair, each outcome has equal probability. Total outcomes = 6 Favorable outcome = 1 (only one face shows 4) So, P(X = 4) = 1/6. Answer: 1/6…

Card 2Discrete Random Variables

Two coins are tossed. Let X be the number of heads. List all values of X and their probabilities.

Answer

Sample space: {HH, HT, TH, TT} X(HH) = 2, X(HT) = 1, X(TH) = 1, X(TT) = 0 So possible values of X: 0, 1, 2 P(X=0) = P(TT) = 1/4 P(X=1) = P(HT or TH) = 2/4 = 1/2 P(X=2) = P(HH) = 1/4 Probability distri…

Card 3Mean of Random Variable

Find the mean of a discrete random variable X with values: -1, 0, 1, 2 and probabilities: 1/8, 3/8, 3/8, 1/8.

Answer

Mean μ = Σ x_i · P(X = x_i) = (-1)(1/8) + (0)(3/8) + (1)(3/8) + (2)(1/8) = -1/8 + 0 + 3/8 + 2/8 = (-1 + 3 + 2)/8 = 4/8 = 1/2 Answer: Mean = 1/2…

Card 4Variance of Random Variable

For the same distribution as above, find the variance.

Answer

We already found μ = 1/2 Variance σ² = Σ x_i² · P(X = x_i) - μ² First compute Σ x_i² · P(X = x_i): = (-1)²(1/8) + (0)²(3/8) + (1)²(3/8) + (2)²(1/8) = 1(1/8) + 0 + 1(3/8) + 4(1/8) = 1/8 + 3/8 + 4/8 = 8…

Card 5Probability Distribution

A random variable X has P(X=1)=k, P(X=2)=2k, P(X=3)=3k. Find k.

Answer

Sum of all probabilities must be 1. So, k + 2k + 3k = 1 ⇒ 6k = 1 ⇒ k = 1/6 Answer: k = 1/6…

Card 6Probability Distribution

X is a discrete random variable with P(X=x) = kx for x=1,2,3,4. Find k.

Answer

Total probability = 1 So, P(1)+P(2)+P(3)+P(4) = k(1) + k(2) + k(3) + k(4) = 10k = 1 ⇒ k = 1/10 Answer: k = 1/10…

Card 7Binomial Distribution

A coin is tossed 5 times. What is the probability of getting exactly 3 heads?

Answer

This is a binomial experiment. n = 5, x = 3, p = 1/2, q = 1/2 P(X=3) = ⁵C₃ (1/2)³ (1/2)² = 10 × (1/8) × (1/4) = 10 × 1/32 = 10/32 = 5/16 Answer: 5/16…

Card 8Binomial Distribution

In 6 throws of a die, find the probability of getting exactly two '6's.

Answer

Binomial: n=6, x=2, p=1/6, q=5/6 P(X=2) = ⁶C₂ (1/6)² (5/6)⁴ = 15 × (1/36) × (625/1296) = 15 × 625 / (36 × 1296) = 9375 / 46656 Simplify: divide numerator and denominator by 3 → 3125 / 15552 Answer: 31…

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Frequently Asked Questions

What are the important topics in Random Variables and Probability Distributions for Telangana Open School (TOSS) Class 12 Mathematics?
Key topics in Random Variables and Probability Distributions include Random Variables and Their Types, Probability Distribution of a Discrete Random Variable, Mean, Variance, and Standard Deviation, Binomial Distribution. Study these first, then practise questions on each for the Telangana Open School (TOSS) Class 12 board exam.
How many flashcards are available for Random Variables and Probability Distributions?
There are 25 flashcards for Random Variables and Probability Distributions covering key definitions, facts and ideas. A few sample cards are shown on this page.
How should I revise Random Variables and Probability Distributions for the Telangana Open School (TOSS) Class 12 board exam?
Learn the core ideas first, then work through the 49 practice questions on Random Variables and Probability Distributions. Revise definitions regularly and use flashcards for quick recall before the exam.

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