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Probability Distributions

Tamil Nadu Board · Class 12 · Mathematics

Flashcards for Probability Distributions — Tamil Nadu Board Class 12 Mathematics. Quick Q&A cards covering key concepts, definitions, and formulas.

45 questions20 flashcards5 concepts

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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 Probability Distributions - Complete Chapter Overview, Problem-Solving Strategy for Probability Distribution Problems, Shows how a random variable maps sample space outcomes to real numbers. These are the concepts Tamil Nadu Board Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Probability Distributions — Tamil Nadu Board Class 12 Mathematics?
Understand the core concepts first, then work through the 45 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.
How many flashcards are available for Probability Distributions?
There are 20 flashcards for Probability Distributions covering key definitions, formulas, and concepts. Use them daily for 10–15 minutes for best results.

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