A random experiment of tossing a die twice is performed. Random variable X on this
sample space is defined to be the sum of two numbers turning up on the toss. Find the
discrete probability distribution for the random variable X and compute the corresponding
mean and standard deviation.
A dictionary contains 80,000 words, of which a certain student knows 30,000. If 100 of the 80,000 words are selected at random from the dictionary without replacement, find the probability:the student knows exactly 40 of the 100 words
To find the probability distribution for the random variable X, we need to consider all possible outcomes of tossing a die twice.
Sample space (S): {1, 2, 3, 4, 5, 6} for each toss
Now, we can find all possible outcomes of X, which is the sum of two numbers turning up on the toss:
When two dice show 1:
X = 1 + 1 = 2
When two dice show 2:
X = 1 + 2 = 3
X = 2 + 1 = 3
When two dice show 3:
X = 1 + 3 = 4
X = 2 + 2 = 4
X = 3 + 1 = 4
When two dice show 4:
X = 1 + 4 = 5
X = 2 + 3 = 5
X = 3 + 2 = 5
X = 4 + 1 = 5
When two dice show 5:
X = 1 + 5 = 6
X = 2 + 4 = 6
X = 3 + 3 = 6
X = 4 + 2 = 6
X = 5 + 1 = 6
When two dice show 6:
X = 1 + 6 = 7
X = 2 + 5 = 7
X = 3 + 4 = 7
X = 4 + 3 = 7
X = 5 + 2 = 7
X = 6 + 1 = 7
Now, let's calculate the probabilities for each outcome:
P(X = 2) = 1/36
P(X = 3) = 2/36 = 1/18
P(X = 4) = 3/36 = 1/12
P(X = 5) = 4/36 = 1/9
P(X = 6) = 5/36
P(X = 7) = 6/36 = 1/6
To calculate mean (μ) and standard deviation (σ), we can use the following formulas:
Mean (μ) = ∑X * P(X)
Standard Deviation (σ) = √∑(X-μ)² * P(X)
Let's calculate:
Mean (μ) = 2 * (1/36) + 3 * (1/18) + 4 * (1/12) + 5 * (1/9) + 6 * (5/36) + 7 * (1/6)
= (2 + 3/2 + 4/3 + 5/2 + 10/3 + 7/6)
= (12/6 + 9/6 + 8/6 + 15/6 + 20/6 + 7/6)
= 71/6
≈ 11.83
Standard Deviation (σ) = √((2-μ)² * (1/36) + (3-μ)² * (1/18) + (4-μ)² * (1/12) + (5-μ)² * (1/9) + (6-μ)² * (5/36) + (7-μ)² * (1/6))
First, substitute the value of μ = 71/6 into the formula.
Standard Deviation (σ) = √((2 - 71/6)² * (1/36) + (3 - 71/6)² * (1/18) + (4 - 71/6)² * (1/12) + (5 - 71/6)² * (1/9) + (6 - 71/6)² * (5/36) + (7 - 71/6)² * (1/6))
Now, calculate the standard deviation (σ) using the above formula. It will be approximately equal to a specific value.
To find the discrete probability distribution for the random variable X, we need to determine the probability of each possible outcome.
Let's start by listing all the possible outcomes when a die is tossed twice:
1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, 1 + 4 = 5, 1 + 5 = 6, 1 + 6 = 7
2 + 1 = 3, 2 + 2 = 4, 2 + 3 = 5, 2 + 4 = 6, 2 + 5 = 7, 2 + 6 = 8
3 + 1 = 4, 3 + 2 = 5, 3 + 3 = 6, 3 + 4 = 7, 3 + 5 = 8, 3 + 6 = 9
4 + 1 = 5, 4 + 2 = 6, 4 + 3 = 7, 4 + 4 = 8, 4 + 5 = 9, 4 + 6 = 10
5 + 1 = 6, 5 + 2 = 7, 5 + 3 = 8, 5 + 4 = 9, 5 + 5 = 10, 5 + 6 = 11
6 + 1 = 7, 6 + 2 = 8, 6 + 3 = 9, 6 + 4 = 10, 6 + 5 = 11, 6 + 6 = 12
Since each outcome has an equal probability of 1/36, we can determine the probability for each sum by counting the number of outcomes that result in that sum.
The probability distribution for X is as follows:
X = 2, P(X = 2) = 1/36
X = 3, P(X = 3) = 2/36
X = 4, P(X = 4) = 3/36
X = 5, P(X = 5) = 4/36
X = 6, P(X = 6) = 5/36
X = 7, P(X = 7) = 6/36
X = 8, P(X = 8) = 5/36
X = 9, P(X = 9) = 4/36
X = 10, P(X = 10) = 3/36
X = 11, P(X = 11) = 2/36
X = 12, P(X = 12) = 1/36
To compute the corresponding mean, we can use the formula:
Mean (μ) = ∑ (x * P(X = x)), where x represents the possible values of X.
Mean (μ) = (2 * 1/36) + (3 * 2/36) + (4 * 3/36) + (5 * 4/36) + (6 * 5/36) + (7 * 6/36) + (8 * 5/36) + (9 * 4/36) + (10 * 3/36) + (11 * 2/36) + (12 * 1/36)
Simplifying the equation gives:
Mean (μ) = 7
The standard deviation (σ) can be calculated using the formula:
Standard Deviation (σ) = √(∑ ((x - μ)^2 * P(X = x)))
Standard Deviation (σ) = √(((2 - 7)^2 * 1/36) + ((3 - 7)^2 * 2/36) + ((4 - 7)^2 * 3/36) + ((5 - 7)^2 * 4/36) + ((6 - 7)^2 * 5/36) + ((7 - 7)^2 * 6/36) + ((8 - 7)^2 * 5/36) + ((9 - 7)^2 * 4/36) + ((10 - 7)^2 * 3/36) + ((11 - 7)^2 * 2/36) + ((12 - 7)^2 * 1/36))
Simplifying the equation gives:
Standard Deviation (σ) ≈ 2.415
Therefore, the mean of the random variable X is 7 and the standard deviation is approximately 2.415.
36 possibilities
number of times sum
1 * 2 = 2
2 * 3 = 6
3 * 4 = 12
4 * 5 = 20
5 * 6 = 30
6 * 7 = 42
5 * 8 = 40
4 * 9 = 36
3 * 10= 30
2 * 11= 22
1 * 12= 12
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36 and 252
mean = 252/36 = 7
go ahead and compute sigma