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Probability Distributions — Flashcards

Tamil Nadu Board · Class 12 · Mathematics

20 flashcards for Probability Distributions (Tamil Nadu Board Class 12 Mathematics) to test yourself on key terms and facts.

45 questions20 flashcards5 formulas & key relations5 concepts

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20 Flashcards·
Discrete Random Variables and PMFDiscrete Random Variables - Dice ProblemsProbability Mass Function - Interval ProbabilitiesCDF to PMF ConversionUnderstanding CDF vs PMFContinuous Probability Density FunctionsPDF from CDF - Uniform DistributionExpected Value - Applications
Card 1Discrete Random Variables and PMF

Two coins are tossed simultaneously. Find the probability mass function for the number of heads occurred.

Answer

Step 1: Identify sample space: S = {TT, TH, HT, HH} Step 2: Let X = number of heads - X(TT) = 0 - X(TH) = 1 - X(HT) = 1 - X(HH) = 2 Step 3: Count frequencies - X = 0: 1 outcome (TT) - X = 1: 2 outco…

Card 2Discrete Random Variables - Dice Problems

A pair of fair dice is rolled once. X denotes the total score. Find P(X = 7) and P(X ≥ 10).

Answer

Step 1: Identify total outcomes - Total outcomes when rolling two dice = 36 Step 2: Find outcomes where X = 7 Pairs that sum to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) Number of favorable outcome…

Card 3Probability Mass Function - Interval Probabilities

The probability mass function is: f(x) = 1/30 for x = 1, 2/30 for x = 2, 6/30 for x = 3, 5/30 for x = 4, 6/30 for x = 5, 10/30 for x = 6. Find P(2 < X < 6) and P(X ≤ 4).

Answer

Step 1: Verify that f(x) is a valid PMF Sum = 1/30 + 2/30 + 6/30 + 5/30 + 6/30 + 10/30 = 30/30 = 1 ✓ Step 2: Calculate P(2 < X < 6) This means X can be 3, 4, or 5 (not including 2 and 6) P(2 < X < 6)…

Card 4CDF to PMF Conversion

Given CDF: F(x) = 0 for x < -2, F(x) = 0.25 for -2 ≤ x < -1, F(x) = 0.60 for -1 ≤ x < 0, F(x) = 0.90 for 0 ≤ x < 1, F(x) = 1 for x ≥ 1. Find the PMF and calculate P(X < 0) and P(X ≥ -1).

Answer

Step 1: Extract PMF from CDF using f(x) = F(x) - F(x⁻) Jump at x = -2: f(-2) = F(-2) - F(-2⁻) = 0.25 - 0 = 0.25 Jump at x = -1: f(-1) = F(-1) - F(-2) = 0.60 - 0.25 = 0.35 Jump at x = 0: f(0) = F(0) - …

Card 5Understanding CDF vs PMF

When do you use cumulative distribution function (CDF) instead of probability mass function (PMF)? Provide an example.

Answer

Use CDF when you need: 1. Probability that X is less than or equal to a specific value: P(X ≤ a) 2. Probability within a range: P(a < X ≤ b) = F(b) - F(a) 3. Probability that X exceeds a value: P(X > …

Card 6Continuous Probability Density Functions

Find the constant C such that f(x) = Cx² for 1 < x < 4 (and 0 otherwise) is a probability density function. Then calculate P(1.5 < X < 3).

Answer

Step 1: Use the condition ∫_{-∞}^{∞} f(x)dx = 1 ∫_{-∞}^{∞} f(x)dx = ∫_1^4 Cx² dx = 1 Step 2: Evaluate the integral C[x³/3]_1^4 = 1 C[(4³/3) - (1³/3)] = 1 C[(64/3) - (1/3)] = 1 C[63/3] = 1 21C = 1 C …

Card 7PDF from CDF - Uniform Distribution

A continuous random variable X has CDF: F(x) = 0 for x < 0, F(x) = x for 0 ≤ x < 1, F(x) = 1 for x ≥ 1. Find the PDF f(x) and calculate P(0.2 ≤ X ≤ 0.7).

Answer

Step 1: Find PDF by differentiating CDF f(x) = dF(x)/dx For x < 0: f(x) = d(0)/dx = 0 For 0 ≤ x < 1: f(x) = d(x)/dx = 1 For x ≥ 1: f(x) = d(1)/dx = 0 Therefore: f(x) = 1 for 0 < x < 1 f(x) = 0 other…

Card 8Expected Value - Applications

Two balls are selected from an urn with 6 white and 4 black balls. You win ₹20 for each black ball and lose ₹10 for each white ball. Find E(X) where X is winning amount.

Answer

Step 1: Identify possible outcomes and X values - 2 black balls: X = 2(20) = ₹40 - 1 black, 1 white: X = 20 - 10 = ₹10 - 2 white balls: X = -2(10) = -₹20 Step 2: Calculate total ways and favorable ou…

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Frequently Asked Questions

What are the important topics in Probability Distributions for Tamil Nadu Board Class 12 Mathematics?
Key topics in Probability Distributions include Random Variables, Discrete Random Variables and Probability Mass Function (PMF), Cumulative Distribution Function (CDF), Continuous Random Variables and Probability Density Function (PDF). Study these first, then practise questions on each for the Tamil Nadu Board Class 12 board exam.
How many flashcards are available for Probability Distributions?
There are 20 flashcards for Probability Distributions covering key definitions, facts and ideas. A few sample cards are shown on this page.
How should I revise Probability Distributions for the Tamil Nadu Board Class 12 board exam?
Learn the core ideas first, then work through the 45 practice questions on Probability Distributions. Revise definitions regularly and use flashcards for quick recall before the exam.

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