A pair of dice is rolled. The random variable x is defined as the sum of the scores obtained. Find the probability function, mathematical expectation and variance



Answer to a math question A pair of dice is rolled. The random variable x is defined as the sum of the scores obtained. Find the probability function, mathematical expectation and variance

Expert avatar
55 Answers
1. The possible sums \(x\) that can be obtained when a pair of dice is rolled range from 2 to 12.
2. Calculate the probability function \( P(X = x) \):

P(X = x) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}

P(X = x) = \begin{cases} \frac{x-1}{36} & x = 2,3,4,5,6,7 \\\frac{13-x}{36} & x = 8,9,10,11,12 \\0 & \text{otherwise}\end{cases}

3. Calculate the mathematical expectation \( E(X) \):

E(X) = \sum_{x=2}^{12} x \cdot P(X = x)

E(X) = \sum_{x=2}^{7} x \cdot \frac{x-1}{36} + \sum_{x=8}^{12} x \cdot \frac{13-x}{36}

4. Simplify the summation to find \( E(X) \):

E(X) = \sum_{x=2}^{7} \frac{x(x-1)}{36} + \sum_{x=8}^{12} \frac{x(13-x)}{36}

E(X) = 7

5. Calculate the variance \( \text{Var}(X) \):

\text{Var}(X) = E(X^2) - [E(X)]^2

6. Calculate \( E(X^2) \):

E(X^2) = \sum_{x=2}^{12} x^2 \cdot P(X = x)

7. Simplify the summation to find \( E(X^2) \) and then \( \text{Var}(X) \):

E(X^2) = \sum_{x=2}^{7} \frac{x^2(x-1)}{36} + \sum_{x=8}^{12} \frac{x^2(13-x)}{36}

\text{Var}(X) = E(X^2) - [E(X)]^2

\text{Var}(X) = \frac{35}{6}

Therefore, the probability function, mathematical expectation, and variance are:

P(X = x) = \begin{cases} \frac{x-1}{36} & x = 2,3,4,5,6,7 \\\frac{13-x}{36} & x = 8,9,10,11,12 \\0 & \text{otherwise}\end{cases}

E(X) = 7

\text{Var}(X) = \frac{35}{6}

Frequently asked questions (FAQs)
Math Question: What is the final factored form of 2x^2 + 5x - 12 using the distributive property?
What is the variance of a data set {1, 3, 4, 6, 8}?
What is the standard deviation of the scores 85, 79, 92, 88, and 90?
New questions in Mathematics
A circle with a 12-inch diameter is folded in half and then folded in half again. What is the area of the resulting shape?
Write 32/25 as a percent
Evaluate limx→∞tan−1(x) using that y=tan−1(x) exactly when x=tan(y) . (Hint: Both tan and tan−1 are continuous!)
The Lenovo company manufactures laptop computers, it is known that for every 60 laptops produced, 54 go on the market with the highest quality standards. If a sample of 15 laptops is taken, calculate the probability that: Exactly 2 are not of high quality
3x+2/2x-1 + 3+x/2x-1 - 3x-2/2x-1
The data set (75, 85, 58, 72, 70, 75) is a random sample from the normal distribution No(µ, σ). Determine a 95% two-sided confidence interval for the mean µ .
224 × (6÷8)
Perpetual annuities are a series of payments whose duration has no end. Explain how can we calculate them, if they have no end?
A company that manufactures personal hygiene items purchases machinery for $220,000 that is considered to last 7 years; it is estimated that at the end of the period it will have a salvage value of $1000. Find: to. The depreciation rate. b. The book value at the end of the sixth year.
Log5 625
It is known that the content of milk that is actually in a bag distributes normally with an average of 900 grams and variance 25 square grams. Suppose that the cost in pesos of a bag of milk is given by 𝐶(𝑥) = { 3800 𝑠𝑖 𝑥 ≤ 890 4500 𝑠𝑖 𝑥 > 890 Find the expected cost.
Let v be the set of all ordered pairs of real numbers and consider the scalar addition and multiplication operations defined by: u+v=(x,y)+(s,t)=(x+s+1,y+t -two) au=a.(x,y)=(ax+a-1,ay-2a+2) It is known that this set with the operations defined above is a vector space. A) calculate u+v is au for u=(-2,3),v=(1,-2) and a=2 B) show that (0,0) #0 Suggestion find a vector W such that u+w=u C) who is the vector -u D) show that axiom A4 holds:-u+u=0
We have received our p&l statement back from accounts. The board has asked for an innovation hub. What items should we prioritise reviewing to decide if we can afford an innovation hub?
find missing measure for triangle area = 48 m square base = 10m heaighy = ? m
A house located within the city limits has a current market value of $325,000 according to a recent appraisal. The assessed value from the last county wide tax valuation is $272,475. The tax rate is $0.36 per hundred for the county and $0.72 per hundred for the city. What is the total annual property tax liability on the property? $2340 $3510 $1962 $2943
We plan to test whether the mean mRNA expression level differs between two strains of yeast, for each of 8,000 genes. We will measure the expression levels of each gene, in n samples of strain 1 and m samples of strain 2. We plan to compute a P-value for each gene, using an unpaired two-sample t-test for each gene (the particular type of test does not matter). a) What are the null hypotheses in these tests (in words)? [2] b) If, in fact, the two strains are identical, how many of these tests do we expect to produce a P-value exceeding 1/4? [2]
Calculate the area of the parallelogram with adjacent vertices (1,4, −2), (−3,1,6) 𝑦 (1, −2,3)
94 divided by 8.75
A grain silo has a height of 8.8m with a 11.4m diameter. If it is filled 0.5% of it's volume, how much grain (m^3) is stored in the silo? (0 decimal places)