Random Variables and Probability Distributions
Telangana Open School (TOSS) · Class 12 · Mathematics
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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…
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…
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…
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…
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…
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…
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…
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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