The random variable X follows normal distribution.P(X<35)=0.2 and P(35<X<45)=0.65.Find mean and standard deviation.
To find the mean and standard deviation of a normal distribution, you need specific information about the distribution. In this case, we are given probabilities for certain ranges of values.
Let's denote the mean of the normal distribution as mu (µ) and the standard deviation as sigma (σ).
Given that P(X < 35) = 0.2, we can interpret it as the cumulative probability of X being less than 35. The cumulative probability of a normal distribution represents the probability that a random variable is less than or equal to a given value.
From this information, we can find the Z-score corresponding to the cumulative probability. The Z-score measures the number of standard deviations a given value is from the mean of the distribution.
Using a standard normal distribution table or a calculator:
P(X < 35) = P(Z < (35 - µ) / σ) = 0.2
By looking up the value in the standard normal distribution table, you can find that the Z-score corresponding to a cumulative probability of 0.2 is approximately -0.84.
Using the Z-score formula:
-0.84 = (35 - µ) / σ
Next, we are given P(35 < X < 45) = 0.65. This represents the probability that X falls within the range of 35 and 45.
To find this probability, we can subtract the cumulative probability of X being less than or equal to 35 from the cumulative probability of X being less than or equal to 45:
P(35 < X < 45) = P( X < 45) - P(X < 35)
Using the Z-score formula, we can express this in terms of Z-scores:
P(35 < X < 45) = (45 - µ) / σ - (35 - µ) / σ= [45 - 35] / σ = 10 / σ
We know that this probability is equal to 0.65. Thus:
10 / σ = 0.65
Now, we have two equations with two unknowns:
-0.84 = (35 - µ) / σ
10 / σ = 0.65
Solving these equations simultaneously will give us the values of µ (the mean) and σ (the standard deviation).