The following table shows the density of the random variable X.
x 1 2 3 4 5 6 7
p(x) 0.03 0.01 0.04 0.3 0.3 0.1 0.01
8
?
A. Find P(8)
B. Find P(3 ¡Ü X ¡Ü 5)
C. Find P(X ¡Ü 4) and P (X < 4). Are the probalities the same?
D. Find F(-3)and F(10)
A. Well, according to the given table, there is no value for X at 8. So, P(8) would be 0. It's like trying to find a unicorn, you won't have much luck.
B. To find P(3 ≤ X ≤ 5), we just need to add up the probabilities for X = 3, X = 4, and X = 5. So, P(3 ≤ X ≤ 5) = 0.04 + 0.3 + 0.3 = 0.64. Who knew math could be this delicious, like a slice of pie?
C. To find P(X ≤ 4), we need to add up the probabilities for X = 1, X = 2, X = 3, and X = 4. So, P(X ≤ 4) = 0.03 + 0.01 + 0.04 + 0.3 = 0.38. Now, let's find P(X < 4). We just need to add up the probabilities for X = 1, X = 2, and X = 3. So, P(X < 4) = 0.03 + 0.01 + 0.04 = 0.08. Are the probabilities the same? Well, no, they are different! It's like comparing the probability of finding a parking spot in a busy city to the probability of finding Waldo in a crowded picture book.
D. Unfortunately, the table doesn't provide any probabilities for X values less than 1 or greater than 7. So, we can't find F(-3) or F(10) from the given information. It's like trying to find treasure in an empty chest or trying to find WiFi signal in a remote desert. It's just not happening!
A. To find P(8), we look at the table and notice that no value of x corresponds to 8, so P(8) is equal to 0.
B. To find P(3 ≤ X ≤ 5), we sum up the probabilities of the values between 3 and 5 inclusive:
P(3 ≤ X ≤ 5) = P(X = 3) + P(X = 4) + P(X = 5) = 0.04 + 0.3 + 0.3 = 0.64
C. To find P(X ≤ 4), we sum up the probabilities of all values less than or equal to 4:
P(X ≤ 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) = 0.03 + 0.01 + 0.04 + 0.3 = 0.38
To find P(X < 4), we sum up the probabilities of all values less than 4:
P(X < 4) = P(X = 1) + P(X = 2) + P(X = 3) = 0.03 + 0.01 + 0.04 = 0.08
The probabilities P(X ≤ 4) and P(X < 4) are different.
D. To find F(-3) and F(10), we need to use the cumulative distribution function (CDF). The CDF at a given x-value is the sum of probabilities of all values less than or equal to that x-value.
F(-3) = P(X ≤ -3) = 0 (since there are no values less than or equal to -3 in the table)
F(10) = P(X ≤ 10) = 1 (since all values in the table are less than or equal to 10)
Therefore, F(-3) = 0 and F(10) = 1.
To answer the questions, we need to use the probabilities listed in the table.
A. To find P(8), we look for the value 8 in the x column. Since the value 8 is not listed in the table, there is no probability associated with it. Therefore, P(8) = 0.
B. To find P(3 ≤ X ≤ 5), we need to sum up the probabilities for the values of X between 3 and 5 (inclusive). From the table, we see that the probabilities for X = 3, 4, and 5 are 0.04, 0.3, and 0.3 respectively. Summing these probabilities gives us P(3 ≤ X ≤ 5) = 0.04 + 0.3 + 0.3 = 0.64.
C. To find P(X ≤ 4), we need to sum up the probabilities for the values of X that are less than or equal to 4. From the table, we can see that the probabilities for X = 1, 2, 3, and 4 are 0.03, 0.01, 0.04, and 0.3 respectively. Summing these probabilities gives us P(X ≤ 4) = 0.03 + 0.01 + 0.04 + 0.3 = 0.38.
To find P(X < 4), we need to sum up the probabilities for the values of X that are strictly less than 4. From the table, we can see that the probabilities for X = 1, 2, and 3 are 0.03, 0.01, and 0.04 respectively. Summing these probabilities gives us P(X < 4) = 0.03 + 0.01 + 0.04 = 0.08.
The probabilities P(X ≤ 4) and P(X < 4) are not the same because when we consider X ≤ 4, we include the probability for X = 4 (0.3), whereas when we consider X < 4, we exclude the probability for X = 4.
D. To find F(-3) and F(10), we need to find the cumulative probability distribution function (CDF). The CDF gives the probability of X being less than or equal to a given value.
Since -3 and 10 are not listed in the table, we need to find the closest values in the table and use the probabilities associated with those values.
The closest value to -3 in the table is 1. The probability for X = 1 is 0.03. Therefore, F(-3) = P(X ≤ -3) ≈ P(X ≤ 1) = 0.03.
Similarly, the closest value to 10 in the table is 7. The probability for X = 7 is 0.01. Therefore, F(10) = P(X ≤ 10) ≈ P(X ≤ 7) = 0.01.
Please note that the approximations are made based on the closest values available in the table.