1. Mean of random variable

Hello guys, we are starting random variables and I know that the mean of a random variable is the function times the probability for all x's (right?), but I am lost with the next two problems. Any help on number 1 would be greatly appreciated, full answer or not. For number 2 I am totally lost.

1. What is the mean of X which has the following density function?
f X (x) = 1/C; for: 0 <= x <= 7
Hint: Find the numerical value of C

2. Consider the probability density function given by:
P X (x) = lambda/2; for: 0<= x <=2
P X (x) = (1 - lambda)/4; for: 2<= x <=6

This is a mixture of two uniform distributions. a)Sketch P X (x) for lambda=0.5. b) Sketch P X (x) for lambda=0.333. c) In a random sample from P X (x), 60% of the values were between 0 and 2. What would be an estimate for the value of lambda?

2. The total probability mass must be one.
So find C such that $\int\limits_0^7 {\frac{1}{C}dx} = 1$.

For #2 do similarly.

3. thanks a lot
could you elaborate a bit on problem #2 please?

Hello guys, we are starting random variables and I know that the mean of a random variable is the function times the probability for all x's (right?), but I am lost with the next two problems. Any help on number 1 would be greatly appreciated, full answer or not. For number 2 I am totally lost.

1. What is the mean of X which has the following density function?
f X (x) = 1/C; for: 0 <= x <= 7
Hint: Find the numerical value of C

2. Consider the probability density function given by:
P X (x) = lambda/2; for: 0<= x <=2
P X (x) = (1 - lambda)/4; for: 2<= x <=6

This is a mixture of two uniform distributions. a)Sketch P X (x) for lambda=0.5. b) Sketch P X (x) for lambda=0.333. c) In a random sample from P X (x), 60% of the values were between 0 and 2. What would be an estimate for the value of lambda?
$\int_0^2 \frac{\lambda}{2} \, dx = 0.6 \Rightarrow \lambda = 0.6$.

5. thank you guys